摘要:

R. BECKER 61 GENERAL PRINCIPLES OF CRYSTAL GROWTH BY PAUL H. EGLI AND S. ZERFOSS Received 18th March, 1949 Certain facts of crystal growth have been observed with sufficient regularity to justify their being regarded as general principles. Some are familiar and will be mentioned only briefly for the sake of coherence. Others, equally significant, have received scant attention. A systematic consideration of this total body of facts should provide the background for extending the theories of crystallization. Classifying the principles of crystal growth in a completely logical fashion is impossible because of the manner in which they are interrelated. Accordingly, some repetition will be necessary and the significance of certain experiments will require discussion in connection with several phases of the general problem.Nucleation Nucleation is discussed as the first phase of the problem because it is the initial step in the overall process of crystallization ; but most of the factors that control nucleation apply in the same manner to growth, and can be discussed more conveniently in that connection. Moreover, the important facts of nucleation are well known and need only to be described briefly.62 GENERAL PRINCIPLES OF CRYSTAL GROWTH (I) The rate of nuclei formation increases with supercooling. Tammann and others1 demonstrated that in melts the nucleation rate reaches a maximum and decreases with further supercooling as diffusion becomes the controlling factor. In solutions such a maximum is difficult to measure and probably does not usually occur.(2) An incubation period is recognized in growth from melts during which nucleation cannot be measured. In solutions, even when seeded, a metastable region of supersaturation is also recognized within which nucleation cannot be measured. Numerous investigators have found, in the phase diagram, well-defined regions with sharp boundaries beyond which nucleation was observable, and it seems reasonable to conclude that under certain con- ditions the rate of nucleation in solution increases extremely rapidly with a small increase in supersaturation. The “ metastable region ” principle is widely employed in the control of industrial crystallization processes and is a useful concept which will be referred to without apology even though the behaviour is more properly described as a rate phenomenon.(3) The extent of the incubation period (or metastable region of super- saturation) can be changed appreciably by slight changes in composition of the system. It is particularly important to note that the metastable region can be greatly increased beyond that of a pure solution by the addition of small amounts of certain additives. This little-explored phenomenon i s well substantiated for numerous compounds and will be further described in subsequent sections dealing with growth factors. (4) The incidence of nucleation depends on the previous history of the system. The evidence supporting the existence of superheatability of nuclei appears overwhelming. The work of Tammann and others1 with organic melts and with metals would seem sufficiently convincing, but the matter is still disputed.The fact that .increasing the amount and time of superheating a system reduces the incidence of nucleation during subsequent supercooling is apparently accepted, but the opponents of the concept of superheated nuclei offer an alternative explanation-that it is insoluble impurities and not nuclei of the principle solid phase which the superheat destroys. This is an important argument in view of the fact that superheat in solids is not predicted in the lattice dynamics of Born or by the theories of melting as recently discussed by Mayer. Accepting the experimental results as evidence of superheated nuclei would also appear- to be somewhat out of harmony with Frenkel’s concept of nuclei formation from embryo. This question is difficult to settle by investigation of solution systems because of the experimental difficulties of observing the early stages of nucleation. Efforts have been made at the Naval Research Laboratory to obtain reliable data by means of heat effects, Tyndall effects, small- angle scattering of X-rays, etc.-all with little success.Qualitatively,. however, it was the general conclusion of the several chemists concerned with growth of numerous crystals from solution that the existence of super- heated nuclei was verified by their experiments on the preparation of saturated solutions. This work is mentioned because in solution systems an explanation based on impurities is extremely unlikely. The reagents were prepared with great care, and in several instances any remaining impurities were in amounts less than those detectable by ordinary analytical techniques.(5) Nucleation is induced by the presence of foreign bodies and by agitation of the system. These well-known facts deserve to be listed in Tammann, Aggregatzustande (Leipzig, 1922). * Frenkel, Kinetic Theory of Liquids (Oxford Univ. Press, 1946).PAUL H. EGLI AND S. ZERFOSS 63 view of the frequent statements that nucleation always occurs for these reasons, or even stronger, that nucleation can never occur without such assistance. Again, these are difficult statements to disprove completely by experiment. Crystal Growth Almost all the phenomena of crystal growth-inclusion of impurities, habit modification, the genesis of twins and flaws-can be resolved into problems of growth rate under certain conditions.Accordingly the growth principles are presented, for the most part, on this basis. There are certain important phenomena, however, for which the rate aspect is not a convenient viewpoint, and which thus necessitate a more complete description. (I) Growth rate increases with increasing supersaturation (or supercooling) and with agitation. The ramifications of these facts are too well known to require elaboration except perhaps noting that diffusion is remarkably constant in all water solutions, and also that normally a moderate amount of agitation is sufficient to eliminate diffusion as a controlling factor. (2) Different faces of a single crystal (under the same degree of super- saturation and agitation) grow at different rates. The rules governing this significant feature of the growth process have been the subject of refinement throughout the history of crystal re~earch.~ In general, crystals possess faces of low indices because of the differential bonding along the few principal directions within the lattice.This idea can be restated in terms of growth toward a minimum free surface energy. (3) The difference in the growth rate of various faces becomes smaller as the overall rate is increased. This long-recognized fact has been demon- strated in a sufficient number of systems and under a sufficient variety of conditions that it can be safely regarded as a general rule. (4) Flawed surfaces grow more rapidly (under the same degree of super- saturation and agitation) than corresponding surfaces without detectable faults.This is meant to apply to twin boundaries, veils, lineage, mosaic structure and presumably any other type of large-order defect. Elaboration of this point is helpful in explaining why defects are propagated and frequently induce additional defects during subsequent growth. At growth rates appreciably below the maximum which can be supported for good growth certain types of flaws lose their rate advantage and may be healed over. ( 5 ) The maximum rate at which good growth can be obtained on a particular surface (the “ critical ” rate) decreases as the size of that surface increases. This highly significant fact has apparently received little attention, but the supporting data appear convincing.Yamamoto demonstrated the phenomenon very clearly with NaCl on a microscopic scale. Investigations at the Naval Research Laboratory, particularly by A. A. Kasper, showed similar results for NH,H,P04 grown under a variety of conditions and the effect has been observed qualitatively in the growth of numerous other crystals. The existence of a “critical” rate dependent on size would appear to lead to the conclusion that in practice there is a limit in size to which a good single crystal of each compound can be grown. Experience would appear to bear this out. The possibility is also predicted that there will be crystals in which zones developed by growth of certain faces will invariably be bad whereas adjoining zones may grow well, and that the volume of poor material could be reduced or eliminated by reducing on the seed the size 3 Wells, Phil.Mag., 1946, 37, 184. Buerger, Amer. Miner., 1947, 32. 593. Yamamoto, Sci. Pafiers. Inst. Physic. Chem. Res., 1939, 35, 228.64 GENERAL PRINCIPLES OF CRYSTAL GROWTH of the face which grows poorly (in NaBrO, use a 1x0-cut seed rather than 1x1-cut seed). It is suggested that this principle must also be more clearly recognized in crystallization theory. Presumably in an ideal system in which growth could be maintained at an infinitely slow rate the limit on size would disappear, but it is also possible that the necessary rate would be below that induced by normal fluctuations at equilibrium. Further implications of this feature of the growth process will be discussed in connection with the general problem of growth from the viewpoint of supersaturation. (6) The critical growth rake in solution systems increases with increasing temperature.This may also be true for any system with two or more components when temperature is a variable. Discussion of this point is necessary because of frequent statements that crystals contain more defects when grown at high temperatures. It is an accepted fact that as the temperature of a crystal increases, whether grown at a high temperature or heated after growth, the number and activity of atomic-scale defects increase. This applies, however, only to vacancies and dislocations of single ions or atoms and does not pertain to large-scale order. In fact, the increased activity with temperature, particularly at the surface, tends to improve the large-scale perfection of the structure during the growth process 2 ; experimentally, it has been found that increasing temperature favours the formation of perfect textures over the formation of spontaneous nuclei or the various types of large-scale flaws.In addition to the foregoing list, there is a rarely mentioned feature of the growth process which deserves considerable discussion. No one has yet offered a satisfactory answer to the basic question of why some com- pounds crystallize readily and others very poorly ; and yet there are some striking facts on which to base such a discussion. The first point to be noted is that in aqueous solution systems compounds which grow readily are all quite soluble. In general, slightly soluble compounds are grown with great difficulty, and highly soluble compounds are grown with great ease.This correlation is far from perfect, however, SO that simple solubility is not a sufficient specification for good crystal growth, and it is necessary to consider the state of association. Some evidence indicates that the critical rate for a given surface increases with increasing association of solute. This is first a statement in harmony with the familiar expression of theory-that the growth rate depends on the difference in the chemical potentials of a particle on the crystal surface and of one in the fluid phase. But the statement also implies something more-namely, that increasing association increases the advantage to formation of a perfect structure relative to the formation of flaws or spontaneous nuclei.The implications of these statements can be more readily discussed in terms of supersaturation than from a strict rate viewpoint. As previously discussed in connection with nucleation, this viewpoint is not a rigorous approach in terms of the kinetics of the rate process, but is a valuable concept for the sake of clarity. The problem then becomes one of determining the range of super- saturation which will induce only perfect growth, the additional degrees of supersaturation which will induce flawed growth of various types and, finally, the degree of supersaturation which will induce spontaneous nuclei. The amount of supersaturation which will induce perfect growth, lineage, etc., is of course dependent on the configuration of each crystal surface availa- ble for growth.For the purposes of this discussion, however, this factor can be neglected by assuming that each crystal has some face which is relatively favourable for growth so that the supersaturation required for that growth is small relative to that required for spontaneous nuclei and is somewhat This also has been demonstrated.PAUL H. EGLI AND S. ZERFOSS 65 less than required to initiate a flaw. On this basis compounds which are difficult to crystallize are those which form spontaneous nuclei with very small degrees of supersaturation so that the range which will induce growth but not flaws is vanishingly small. Easily grown crystals are those for which nuclei form only with considerable supersaturation so that there is a large range in which only perfect growth is obtained.Returning to the original assertion that ease of growth increases with increasing association of the solute, it is desirable to consider the supporting evidence from the supersaturation viewpoint. Quantitative data are difficult to obtain because of the difficulty of measuring and rigorously describing the quality of a crystal, but even more because of the lack of quantitative data regarding the amount of association of various salts in highly concen- trated solutions. Acceptable evidence is thus by necessity limited to obvious gross effects, but several compounds can be discussed for which the growth behaviour clearly supports the viewpoint expressed. NaCl is a typical example of a compound which is soluble but highly dissociated in solution, and experience indicates that NaCl is virtually impossible to grow into a perfect crystal at any reasonable rate from pure solution.It forms copious nuclei with very small degrees of supersaturation. HIO, is typical of compounds known to be highly associated in solution. Nuclei form only with considerable supersaturation, and large, perfect crystals are easily grown. Also typical of the associated solutes which support supersaturation and form crystals readily are a large number of hydrated salts. Additional significant evidence was demonstrated by growing benzil from water solution and various organic solvents. Growth was difficult in every solvent except benzene ; from this related environment nuclei were formed only with considerable supersaturation and growth was excellent.Thus all the available evidence appears to be in harmony, but additional data are desirable. Significant information can be derived from the remarkable effect of small concentrations of foreign ions. The growth of several dozens of compounds has been markedly improved by the addition of “ impurities” to the solution. A typical case is NaCl which grows with great difficulty from pure solution but which grows readily from a solution containing Pb. The obvious effect is to greatly increase the range of supersaturation within which spontaneous nuclei do not form. Yamamoto obtained data for several compounds and this work has been extended a t the Naval Research Labora- tory to many solution systenrs.The same phenomenon has also been demonstrated there in the growth of alkali halides from a melt and has been reported by one competent investigator to apply in growth by flame fusion. The rules for selecting the most effective additive are not yet entirely obvious. Heavy metal, multivalent ions in concentrations of less than 0.01 mol-% are frequently the best choice, though small concentrations of anything that, if present in larger amounts, would modify the habit appear to be helpful. If concentrations beyond the optimum are used, the habit is modified, flaws are induced and spontaneous nuclei occur more readily than from pure solution. When properly used, however, this is a very powerful tool. In one case, for example, it made possible the formation of a compound which cannot be precipitated from pure solution ; K,MnCl, can be formed only by the addition of Pb++ to the solution. In a practical sense, the phenomenon is extremely valuable for increasing the efficiency of growth processes for single crystals and has in addition promising applications in industrial crystallization of fine chemicals.From a scientific viewpoint it promises to contribute valuable clues to the overall problem of nucleation and growth. Space does not permit thoroughly justifying the choice of the foregoing C66 GENERAL DISCUSSION factors as being the most significant to interpretation of the overall problem. Obviously many interesting facts have been completely neglected. Habit modification, oriented overgrowths , inclusion of impurities and similar fields of research contribute valuable information but for the most part appear to be less directly necessary for consideration as part of the primary process of crystallization. The points which are discussed can hardly be ignored in even a qualitative theory that is to be of any value. The concept of a critical growth rate dependent on the size of the surface and the extent of association in the fluid phase, for example, appears to be a basic factor in the process. It is hoped that discussion in this manner of a rather loose body of facts may point out more clearly than would a neat mathematical expression the status of our present knowledge and the direction most profitable for future research. Crystal Section, Naval Research Laboratory, Washington, D.C.

ISSN:0366-9033    

DOI:10.1039/DF9490500061

出版商:RSC    

年代:1949

数据来源: RSC

摘要:

66 GENERAL DISCUSSION GENERAL DISCUSSION Dr. F. C, Frank (Bristol) said : In putting forward a structure-sensitive theory of crystal growth I can draw an analogy with the theory of the strength of solids. This is in practice a t least a hundred times less than their theoretical strength. So also crystals show a “ growth resistance ” which is a hundred times less than their theoretical growth resistance. We owe to Volmer, to Farkas and to Becker and Doring the recognition and rather difficult calculation of this theoretical growth resistance of a perfect crystal. It appears from their work that new deposition will not occur a t an observable rate on a completed habit-face of a crystal unless the supersaturation exceeds a substantial critical amount. This is of the order 50 yo for a typical crystal growing from the vapour-greatly in excess of any critical supersaturation for growth which has been observed.We lack data for quantitative calculation of crystals, particularly ionic crystals, growing from solution, but every small crystal which grows in polyhedral shape demonstrates the existence of a substantial “ growth resistance.” The transport of material to a sufficiently small crystal in solution is governed by the diffusion equation. The solution of this differential equation will be uniquely defined if either the concentration or the flux a t the boundary is defined. There is no degree of freedom left to satisfy boundary conditions in terms of both. The growth of the crystal determines a boundary condition in the flux, and the concentration a t the boundary is determined accordingly, and is necessarily non-uniform over the surface of a polyhedral crystal.The flux and concentration can only both be uniform at the surface of a sphere. The surface supersaturation being non- uniform and nowhere negative is at places substantial ; namely, a t the corners, where it reaches several per cent. in the experiments of Humphreys-Owen and Bunn. Thus the very fact that a t least some crystals will grow polyhedrally without becoming dendritic shows that they have in places a substantial “ growth resistance ” : but at the same time the maximum supersaturation such crystals will withstand all over, without growth occurring somewhere, appears experi- mentally, so far, to be less than I yo. Dr. S. P.F. Humphreys-Owen (London) said : Dr. Franksays thatnucleation a t a dislocation, or system of dislocations, is capable of undergoing discontinuous changes. But why should one system be in some way peculiar in that it produces the Nernst mode of growth, i.e., with equilibrium concentration a t the face centre ? Other systems appear to produce rates of growth which are less, andGENERAL DISCUSSION 67 which are associated with a concentration at the face centre which is above the equilibrium value. This indicates the onset of a resistance to growth. Again, how can a system of dislocations, which, according to Dr. Frank, has a consider- able degree of persistence, permit the complete stoppage of growth which is sometimes observed in high supersaturation ? Theories such as Dr.Frank’s, which provide the crystal with a means of overcoming the difficulty of growth of a complete surface, appear to go too far in the other direction; they make growth too easy. Dr. F. C. Frank (Bristol) said : The rate of growth, once in the range of conditions a t which growth proceeds steadily, is very largely independent of the density of dislocations. To see why this is so, we must first consider the effect of a single dislocation. The attached growth front (terrace edge or step-line) winds itself up into a rotating spiral (Fig. 5). A pair of dislocations emit growth fronts in the form of closed rings with about the same spacing, provided the supersaturation is sufficient for them to be effective at all (Fig. 6). Now, growth Y FIG. 5.-Spiral growth front attached to a single dislocation.FIG. 6.-Development of growth fronts in closed loops from a pair of dislocations. fronts differ from wave fronts in the fact that when they meet they annihilate. Hence if two different dislocations or groups of dislocations in the same crystal face are equally active in emitting growth fronts, the number of growth fronts passing any point in the face is the same as if either were active alone. A group of dislocations is as active as its most active member, and the growth rate of the whole face i s determined by its most active dislocation group. All other groups yield within a short while to the domination of this most active group, and merely pass on the growth fronts received from it with a slight delay and in slightly modified form.The members of a group are nevertheless able to stimulate each other to somewhat enhanced activity. This is because a concave growth front, which is formed each time a pair of fronts meet, travels faster than normal, and so “ helps the growth spiral round.” This stimulation occurs at a super- saturation somewhat above the critical supersaturation for the group, below which they are inactive. If the supersaturation is increased considerably above this, the component dislocations then behave as though independent of each other ; but this may be the right supersaturation for stimulation in some other group, which will then become dominant. An exact calculation is not easy, but it does not seem likely that this “ stimulation ” will make a much larger difference than a factor of 2.68 GENERAL DISCUSSION Thus, at least for order of magnitude, the problem of calculating the growth rate of a face with any number of dislocations can be reduced to that of calculating the rate of growth based on a single dislocation : which is simply the rate of generation of fresh turns of the growth front spiral multiplied by the thickness of a molecular layer.As explained in my paper, this rate is proportional to uU,/E,, where a, is proportional to and I , inversely proportional to ol, the super- saturation at the crystal surface in the neighbourhood of the dislocation or dominant dislocation group. Let us say the growth rate is w = AoI2. But the controllable supersaturation o, is that a t some point away from the crystal, separated from i t by a diffusion barrier, as a result of which there is a drop in supersaturation (G, - al) proportional to the growth rate : say, ( oo - G,) = Bw.In consequence a growth rate establishes itself such that Briefly, the growth rate is proportional to the square of the supersaturation when this is small; but at high supersaturation the diffusion barrier takes ABZW = ABo, + &(I - 1 / I + 4ABo,). 1 FIG. 7.-Crystal growth rate as a function of supersaturation (with simple diffusion barrier) :- (a) for a single dislocation. ( b J , (b,) . . . for variously spaced pairs or other groups of dislocations. (c) Resultant growth rate curve when all of the groups ( b J , ( b 2 ) . . . are present. control and ‘‘ linearizes ” the growth rate. This is represented by curve (a) in Fig.7. Here we have assumed the simplest possible sort of diffusion barrier. In growth from a dilute vapour we actually have a more complex situation, with surface diffusion and molecular transport through the gas in ‘‘ series-parallel ” connection. This problem has been treated by Cabrera and leads to a curve in which the transition from the quadratic to the linear law of growtb rates involves a region of reversed curvature. Now, if there are various dislocation groups in the crystal face, which would separately produce growth rate curves ( b J , (b,) . . ,, the resultant growth rate curve is G made up of the curves of the groups dominant at various supersaturations. Change of supersaturation can change the dominant group, thus changing the centre of the growth pyramid ; but the growth rate is not greatly dependent on the dislocation structure once the supersaturation exceeds the critical super- saturation of the least growth-resistant dislocation group.In further discussion with Dr. Cabrera and Mr. Burton, since the meeting, we have arrived at the conclusion that there is no existing experimental evidence of a critical supersaturation which necessarily indicates that dislocations are close together : the quadratic growth law which applies a t small supersaturation when dislocations are far apart suffices to explain the existence of a supersaturation below which the growth rate is too small to be measured by the techniques which have been applied, as in the experiments of Volmer and Schultze, and of Nitschmarin and Spangenberg.GENERAL DISCUSSION 69 Prof. I .N. Stranski (Berlin) said : Hitherto i t has been accepted that in (microscopically) visible crystals a spontaneous alteration of form is not possible in a state of thermal equilibrium. The reason for this has been recently seen to lie not primarily in the smallness of the relative vapour pressure differences between the various crystal faces, but rather in the fact that the building-up of new lattice planes is connected with the work of formation of two-dimensional nuclei, which according to Volmer tends to infinity with decrease of saturation. In the experiments on urotropine the work of formation of two-dimensional nuclei is considerably decreased by the presence of re-entrant edges. In the accompanying Figure three types of re-entrant edges are seen to be active: (i) Mono-crystalline re-entrant edges formed by incompletely grown lattice planes, x.(ii) Poly-crystalline re-entrant edges formed by contact between two differ- ently oriented individual crystals, y . (iii) Heterogeneous re-entrant edges formed by participation of the crystal substrate. According to experiments by Honigmann, the mono-crystalline re-entrant edges x are particularly effective. The work of nucleus formation can only be purely one-dimensional at such defects, that is, even for very small super- saturations it remains negligible.1 The relative vapour pressure difference APjP has been estimated as I O - ~ to I O - ~ , and for a fraction 8 of coverage by the adsorbed layer, for which the comparison is particularly valid, very reasonable values (very much less than I) are obtained.In all these experiments the crystals of urotropine are surrounded only by their own vapour. The admixture of foreign gases increases the times for the transfer of matter enormously. Dr. H. K. Hardy (Stoke Poges) said : If, as Dr. Frank has indicated, crystals grow by " winding themselves up " with screw dislocations, would i t be correct to speak of their solution as an " unwinding " process ? I am prompted to ask this because there was a paper by Bloch, Brings and Kuhn in which the rate of solution of a crystal was taken as proportional to the edge length. On this hypothesis smaller crystals melted more slowly than large ones and this was taken as the source of undissolved crystal nuclei a t temperatures not far from the melting point.If, instead of edge length, we substitute effective dislocation length as the criterion for the rate of solution, a stage will eventually be reached during solution (or melting) at which this is reduced to zero but at which a very small crystal fragment remains. This might be expected to have considerable stability and would form a source of crystal nuclei on subsequent cooling. I would now like to mention some phenomena connected with precipitation in solid solutions since, by quenching from a high temperature, the effects of very high degrees of supersaturation can be studied when the alloy is allowed to 1 See, for example, Stranski and Kaischew, Ann. Physik, 1935, 23, 330. * This Discussion. 3 Bloch, Brings and Kuhn, Z.physik. Chem. B, 1931, 12, 415.70 GENERAL DISCUSSION decompose at a lower temperature. The most convenient starting point is the free energy-composition curve (Fig. I). The equilibrium phases have the compositions given by the common tangent. Between the points Y-Y, the curve is concave to the composition axis and in this region 3 2 F / 3 ~ 2 has a negative value. In agreement with diffusion theory, fluctuation theory and the shape of the curve, we should expect that an alloy quenched to this region would show segregation of like atoms and possibly pre-precipitation effects associated with this. This is the case in the Cu,FeNi, alloy investigated by Daniels 4 in which variation of the ageing temperature changed first the degree of segregation and then the distance between segregates.F (a) The free energy composition curve at TI for the hypothetical equilibrium diagram in Fig. ( b ) (b) The supersaturated solid solution has been divided into two regions corresponding to the inflections on the free energy composition curve and positive and negative values of 32F/3xa. In general, the solute atoms collect on preferred planes as " platelets " or even as '' string let^,"^ but in some alloys the effects are more complicated than can be predicted by simple segregation. For example, in Al-Cu alloys this segre- gation requires nucleation, there being a critical nucleus size dependent on the ageing temperature. If we consider the Al-4 % Cu alloy, the segregation of solute atoms at room temperature after quenching from 530" C leads to an increase in hardness (Fig.2). If the alloy is now raised to zooo C for a few minutes, 4 Daniels, Proc. Physic. SOC., 1948, 192, 575. 6 Geisler and Hill, Acta Cryst., 1948, I, 238.GENERAL DISCUSSION 71 the segregates are dispersed and i t reverts to its hardness after quenching. It will then re-age as before and the process can be repeated for a t least 10 times and probably indefinitely (Fig. 2 ) . The segregates (or Guinier-Preston zones as they are called) formed a t room temperature are below the critical size for their growth at zooo C. This means that the free energy of such zones, even though not a separate phase, possesses a strain energy term proportional to their surface area, and hence their free energy will show a maximum when plotted against their radius (Fig.3 ) . 16 Y =2 I/ =3 1 A9einq hme uf room Fernperafure FIG. 2.-Behaviour of Al-4 :,(, Cu alloy on natural ageing following quenching a-&,, treated 5 min. 200' C to disperse the segregates of copper atoms b,-c,, etc., which re-establishes its ability to natural age cl-b,, etc. , Uns a rura lid so lu hon I I 1 I F ,I / FIG. 3.-Variation of Z F with size of nucleus. Both Becker and Borelius have put forward theories to account for the rates of precipitation. Fig. 4 gives experimental curves on Pb-Sn alloys, where the logarithm of the time for half the resistance change associated with precipita- tion has been plotted against the reciprocal of the isothermal precipitating temperature in O K (the thick lines in Fig.4). The arrows mark the temperature, calculated by Borelius 9 at which a2F/3x2 changes sign, at higher temperatures ; where this is positive the rate of precipitation slows down. In Becker's theory 7 the rate of precipitation per unit volume of untransformed matrix is proportional to e-(Q + A)/RT, where Q is the activation energy for diffusion and A the activation energy for nucleation. I have applied this to the experimental results given in Fig. 4. The free energy terms needed were calculated using the methods of Borelius 9 10 and comparing them with Becker's equations.' By this means the activation energy was calculated for the initial precipitation. This leads to the dotted Becker, 2. Metallkunde, 1937, 29 (B), 245. Becker, Ann. Physik., 1938, 32, 128.8 Borelius, Ann. Physik., 1938, 33, 517. 9 Borelius, ArRiv. Mat. Fysik., 1945, 32 (I). 10 Borelius, Lams and Ohlsson, Avkiv Mat. Ast. Fysih. A , 1944, 31 (10).72 GENERAL DISCUSSION curves in Fig. 4. The agreement with experiment is not unreasonable in view of the simplifying assumptions which were made. If a correction were made for the decrease in rate of nucleation due to change in degree of supersaturation during precipitation, the theoretical curves would be moved to the right, i.e., closer to the experimental values. I hope to publish a more detailed account of this work on precipitation in terms of the free energy composition curve a t a later date. / / I I \ \ \ \ 1 2 3 4 5 6 ' I log t (sec.) t = time to half resistance change.* - temperature for change from positive to negative value of 32F/3xa. Full lines experimental, dotted lines calculated. FIG. 4.-Comparison of experimental curves for log time to half the resistance change of Pb-Sn alloys on ageing, plotted against+, and curves calculated from nucleation theory based on the original concentrations. Q = 10,400 cal./g.-atom. Dr. F. C. Frank (Bristol) (communicated) : Dissolution on a habit-face of a crystal with protected edges, a t moderate subsaturation, should proceed in a manner closely equivalent to growth on fully developed habit-faces, i.e. , by unbuilding a t molecular terrace lines ending on screw dislocations, and there should be a critical subsaturation for dissolution, equal to the critical super- saturation for growth.But if the crystal edge is exposed to attack, it is a permanent source of terraces, and the dislocations then play no essential part in the steady-state dissolution process. Terraces attached to dislocations wouldGENERAL DISCUSSION 73 be involved in the initial attack, and would, I think, upset the proportionality between rate of attack and length of crystal edge ; which could in any case only apply to initial conditions, since crystals in dissolution rapidly cease to be simple polyhedra and become bodies bounded by curved or rough surfaces, open to attack all over. The rate of steady-state dissolution is governed simply by transport of heat or matter. It is theoretically possible to dissolve a crystal until the remaining fragment is undislocated, and, if now submitted to moderate supersaturation, can only grow out to the circumscribing polyhedron of habit-faces and then stop growing.I doubt whether this has practical importance, or will be easy to observe. The paper of Bloch, Brings and Kuhn quoted is interesting but illogical. There is no reason why the edge-row (d in their Fig. I) should be less easily attacked than the step-row ( b ) , or the corner atoms of the crystal less easily than those a t " reproducible points " (c). Actually, I know of no real evidence of any portion of a solid failing to melt above its melting point, and believe that " persistent nuclei '' in metal melts are to be explained in a relatively trivial way, as foreign matter, e.g., oxide : the surmise of Horn and Masing l1 for antimony, that the impurity gradually dissolves in the melt, but is precipitated out on solidification, accounts for the details of behaviour, and may well apply in other cases.Mr. W. K. Burton and Dr. N. Cabrera (Bristol) (conznzunicated) : With reference to the theory of crystal growth from the vapour, when there is an adsorbed layer of molecules of high mobility, the present situation can be summarized in the following way. Let D be the diffusion coefficient of the adsorbed atoms and z their mean life on the surface (mean time between their condensation from the vapour and their evaporation again into the vapour). Then the mean displacement of adsorbed atoms is ~ Z = 2/07 = alp, where a is the interatomic distance and p is defined by formula ( 3 ) in Part I1 of our paper.For materials of low sublimation energy like iodine, we expect x' to be of the order of 102 interatomic distances, a t room temperature. It will be much higher for metals. If x' is bigger than the mean distance x, between Kossel-Stranski-Frenkel kinks (places where condensation into the body of the crystal is easy), then it can be shown that the rate of growth of the crystal is proportional to the super- saturation B = 0: - I (formula (6) in Part I1 of our paper). This formula is in good agreement with the linear law observed by Volmer and Schultze12 on iodine, phosphorus and naphthalene at oo C. These authors observed a linear law in all cases, but for iodine the actual rate of growth is smaller than the linear law, below supersaturations of the order I O - ~ .The problem is to explain why there seems to be always such a high concentra- tion of kinks. It is now quite clear that the concentration of kinks in a perfect habit face will be negligible, unless the supersaturation is very high (see our papers), and also that the only explanation for the observed growth a t low supersaturations is the fact that the crystals are imperfect and the screw disloca- tions terminating in the surface (see Frank's paper) provide the required steps with a high concentration of kinks. The step between two dislocations of different sign will contribute to the growth if the distance d between them is bigger than the size I , of the critical two-dimensional nucleus, which is of the order a/o. If N is the number of dislocations per cm.2, then d " - ' / a .Now from the point of view of the rate of growth we can distinguish two cases : either the distances d between dislocations are bigger than the mean displacement x defined above, or they are smaller. I n the first case, d > F, we expect two different critical supersaturations. Increasing the supersaturation B from zero, there will be a first range where no observable growth will occur, up to B = cl N a/d. A t this first critical supersaturation u1 the growth will start more or less suddenly (see our paper, Part 11). t s l will be different from one crystal to another, and will l1 Horn and Masing, 2. EEektrochem., 1940, 46, 109. l2 Volmer and Schultze, 2. 9hysik. Chew. A , 1931, 156, I. C*74 GENERAL DISCUSSION be observable if it is of the order of, or bigger than, IO-~.Actually o1 - a/d -aN’/a, therefore it will be observable if N is bigger than I O ~ O . When o > ol, Frank has shown that a dislocation or small group of dislocations will be the centre of a “ growth pyramid,” the distance between different step-lines of the pyramid being of the order of I , or a/o. When o < og N a/;, the rate of growth of the surface is more or less proportional to 02 and is smaller than that given by the h e a r law. Finally, when G > a2, the parabolic law goes over to the linear law, because the distance between step-lines of the pyramids is smaller than x’ and we are under the conditions where the linear law holds. The second critical supersaturation og, defined by o2 N a/?, will be the same for all crystals, x’ being independent of the imperfection of the crystal.This seems to be the case in the experiments on iodine by Volmer and S c h u l t ~ e . ~ ~ Assuming, then, that the critical supersaturation observed in their experiments corresponds to og, one can deduce the value x’ N 102a, which agrees with the expected value. The rate of growth for o1 c < og will depend essentially on the distribution of dislocations. FIG. I . In the second case, x’ > d, the first critical supersaturation only appears. We should expect the rate of growth to go over to the linear law rather suddenly (as represented by the dotted curve in Fig. I) and the critical supersaturation to be different from one crystal to another. This situation would occur probably in metals, provided the crystal contained a number of dislocations high enough for the critical supersaturation to be observable.Prof. I . N. Stranski (Berlin) said : Foreign (just as lattice) adsorbed molecules influence the growth and dissolution processes in two ways. Firstly, they favour (i.e., catalyze) the elementary processes by lowering the amount of energy required. In this connection we may consider the simple electrostatic model of the ionic crystal. To remove one adsorbed ion from a (001) face of a sodium chloride lattice an energy ‘pl = 0.06601 (in e8/ro units) is required. To remove a single ion which is part of the same face an energy ‘p2 = 1.68155 is necessary. To remove an adsorbed ion together with the ion of opposite charge (directly beneath it) an energy ‘plj2 = 0.74757 will be needed.(This latter is the energy of separation of a molecule from the crystal boundary, involving only half-crystal forces.) The first process is naturally the most frequent. The third process can, however, occur much more frequently than the second, which is the relevant point here. The adsorbed foreign molecules, however, retard the diffusion in the adsorption layer itself and in this way they may easily outweigh the first effect.GENERAL DISCUSSION 75 Mr. R. S . Bradley (Leeds) (communicated) : A possible experimental test of Prof. Stranski’s theory of surface polarization is the measurement of the heats of adsorption of the inert gases on ionic crystals, in comparison with the values calculated by the application of Prof. Stranski’s theory.It largely coincides with the views of Honigmann and myself. In our preliminary com- munication l3 we also draw attention to the close connection between the processes occurring on the urotropine crystal with the growth-processes for twins, where likewise a considerable reduction in the work of formation of the two-dimensional nuclei on the boundary lines must be anticipated. We have also considered attributing the unsuccessful experiments of Volmer and Schultze (naphthalene, phosphorus) to the presence of re-entrant edges. Dr. J. L. Amorbs (BarceZona) (communicated) : I t is known from the so-called Donnay-Harker’s law that the equilibrium form of crystals depends upon the real symmetry of the crystal, i.e., the group symmetry. This law has been criticized by me during the last few years.However, in my opinion the law remains valid in all respects because the crystal habit is directly related to the crystal structure. The relation between structure and both growth and final form of the crystal appears to be clear when the stability of crystal body is realized. That stability is reached when the crystal faces are stable and the latter can only be stable when the co-ordination polyhedron or the close-packing of molecules is easily obtained . Therefore the crystal growth must take place on those faces where either the completion of the co-ordination or the close-packing of molecules can be more easily obtained. As the crystal structure determines the symmetry of the cell, the crystal habit is likely to be a function of the space group elements of sym- metry.Rearing in mind the above, it should be interesting to know the actual relationship between Stranski’s theory and the Donnay-Harker’s law. Dr. C. W. Bunn (I.C.I., PZastics) said: The suggestion in Frank’s paper that dislocations play a dominant part in crystal growth from vapour or solution arises from the conclusion that a perfect crystal bounded by low-index surfaces probably would not grow a t all unless the supersaturation of the vapour or solution were very high. But the fact that crystals do grow at quite moderate supersaturations might be explained in two ways-either by assuming the presence of dislocations, or by assuming that the surfaces are not low-index surfaces, The observations described by myself and Emmett in this Discussion show that on many crystals growing rapidly from solution deposition takes place on the edges of spreading layers, and that these edges are, on the scale which can be observed in the optical microscope, high-index surfaces.If all the deposition surfaces are high-index surfaces on the molecular or ionic scale, there would appear to be no need to invoke dislocations to explain the continuance of growth, because even a crystal of perfect structure will grow readily if it has high-index surfaces. Our observations suggest that the central problem of crystal growth is the study of the factors which maintain high-index surfaces -the factors which determine that molecules are deposited in such a way that the new surface is again a high-index surface. There is a tendency for high-index surfaces to “ heal ”-i.e., for depositing molecules to form a low-index surface ; what i s it that prevents complete healing ? Our observations also show that layers usually spread outwards from the centres of crystal faces.Is there any reason to suppose that dislocations would occur preferentially a t face centres ? The electron microscope photograph of Wyckoff’s 14 to which we refer in our paper (a protein crystal grown from aqueous solution) does not show a screw dislocation, or indeed any central struc- tural imperfection in the top layer. It is true that this is not a photograph of an actually growing crystal, but of a crystal whose growth has been arrested, so it must be regarded as suggestive rather than conclusive, for we do not Prof.I . N. Stranski (Berlin) said : Frank’s idea is signihcant. l3 Stranski et at., Naturwiss., 1948, 35, 156. 14 Wyckoff, Acta Cryst., 1948, I, 277.76 GENERAL DISCUSSION know what might have happened after growth ceased. But it corresponds so exactly to what has been observed dynamically on a larger scale that it is difficult to resist the impression that it is a molecular growth-picture, and theref ore deserves consideration. The screw dislocation has been introduced because it is self-perpetuating ; but it seems to me a rather special sort of imperfection, and I cannot visualize how it arises. I should feel happier about an imperfection theory based on simple strains, cracks and dislocations arising continuously and spontaneously perhaps as a result of thermal strains originating in the flow of heat of crystal- lization and depending on previous growth rates.Dr. F. C. Frank (Bristol) (partly communicated) : Prof. Stranski says, rightly, that growth at low supersaturation can occur on uncompleted molecular layers : but then, on a perfect crystal, uncompleted layers must finally become completed ones. From discussion with others, i t appears necessary to say that also in other language : that high index surfaces grow readily, but thereby grow out, leaving low index surfaces. The essential importance of dislocations is that they prevent this happening, enabling the uncompleted layers or high index surfaces to persist. If Wyckoff’s Fig. 7, shown by Dr. Bunn, were a picture of a crystal growing from solution of low supersaturation, I should be rather worried.I t is not : from Wyckoff’s own description, it is a preparation made by smearing the crystalline pellet (formed by ultracentrifugation) on a glass surface. I should like to reply to two more points raised by Dr. Bunn and one raised by Dr. Humphreys-Owen. Firstly, is there any reason to suppose dislocations would occur preferentially a t face centres ? Yes, dislocation lines are effectively under tension. Given the chance they will usually make themselves as short as possible, and this will, as a rule, overcome any tendency of the dislocation line to adhere to a preferred direction in the crystal, when these requirements are in conflict. Hence, they will be expected to grow out more or less normally to the surface a t which they emerge. Those of early origin will then be found near the foot of the normal drawn from the crystal seed to the growing face, and therefore usually somewhere near the centre of the face, a t least in regular crystals.Dislocations of later origin may be anywhere. The observations quoted imply that in these (incidentally, small) crystals the dislocations are few and the probability of fresh generation of dislocations small. Secondly, Dr. Bunn says “ the screw dislocation has been introduced because i t is self-perpetuating.” This must be corrected. The dislocation was “ introduced ” from the theory of continuum elasticity into the theory of real crystalline solids about 15 years ago to account for their plastic deformation, and about 10 years ago as the element of misfit into which various derangements of a crystal (such as “ mosaic boundaries ”) can be resolved.Each dislocation is characterized by a “ Burgers vector,” belonging to a limited family of crystallographic vectors. The idealized “ screw dislocation ” is that in which the dislocation line lies parallel to its Burgers vector ; but if this vector is not parallel to the crystal face a t which the dislocation line terminates, the dislocation partakes of screw character, and provides a self-perpetuating terrace on the crystal face. If disloca- tions are present, it is only in certain special arrangements, for the maintenance of which there is no discoverable reason, that they could fail to possess screw terminations. Of course, dislocation theory will be in a more satisfactory state when a complete account of their genesis has been given.But 1 did not omit to discuss the matter in my paper. Humphreys-Owen raises a very significant point : that the growth-resistance either a t a crystal corner, or of a whole face which has ceased growing, is remark- ably high. By itself this could mean either that the dislocations were very close together in these areas or that they were absent. It is very much easier to under- stand the erratic nature of the growth of these small crystals on the latter assumption, which is also consistent with the evidence that the growing point is often near the centre of the face. In fact, when the growing point is not near the face centre of a small crystal, there is a strong likelihood of the production of a “ hopper crystal.” For then the diffusion field corresponding to uniformGENERAL DISCUSSION 77 deposition will result in a smaller supersaturation at the face centre than that at the growing point.A certain reduction, by a factor of the order 100, can be compensated by the growth fronts crowding together in the region of low super- saturation, but this compensation is definitely limited, whereas the fall in super- saturation which would accompany uniform deposition might even be to a negative value. In such cases deposition cannot be uniform but will be less near the centre of the face. The result will be a hopper crystal. This conclusion applies to small crystals, or those grown in still media, or grown fast. With larger crystals or good stirring the variation in concentration across a face will be much smaller ; on the other hand, when the growth rate is fast the amount of variation in super- saturation which can be compensated by a change in the spacing between step- lines is smaller.Perhaps I should summarize : the dislocation structure of a crystal has very little effect on its rate of growth (provided it permits it to grow at all) if the super- saturation is the same all over the crystal. The latter is not the case with small crystals, or crystals in a still medium. The dislocation density probably varies greatly, from substance to substance and with the conditions of crystal growth and treatment, but we can learn very little about i t from simple measurements of growth rate, or even from the shape and surface topography in growth under usual conditions : it might be anything from I per crystal face to 1010 or more per square centimetre.But there is evidence in the experiments of Bunn, and of Humphreys-Owen, that i t is indeed much nearer to I in some of the small crystals they have studied. In that case, with a variation of concentration across the crystal face, which must exist when diffusion rules, the growth rate can be determined by the location of dislocations in the face. Dr. U. R. Evans (Cambridge) (communicated) : Three important growth forms (dendritic, concentric and allotriomorphic) , mentioned by several speakers, deserve closer consideration. DENDRITIC FoRMS.-Mott attributes dendritic growth to the circumstance that the tip of an advancing needle provides a spot favourable for the replenish- ment of material or for the dissipation of heat.This widely held view that growth is favoured by a sharp point seems a t first sight to conflict with the demonstration of Bunn and Emmett l6 that, under certain circumstances, deposi- tion occurs preferentially a t the centres of the faces-that is, at a maximum distance from the sharpest points on the crystal ; it is even suggested that the replenishment of material a t face centres proceeds more readily than a t crystal corners. The apparent discrepancy may be connected with the fact that, as Bunn himself points out, his theory neglects the growth of the crystal ; this assumption, which he describes as unrealistic, is doubtless permissible for many purposes, but in dendritic growth the movement of the tip of the dendrite is probably an essential feature of the mechanism.The impoverishment of a solution, which must always be expected during the advance of a flat face, may be avoided when a fine needle is pushing out into ever fresh regions. Such a mechanism would lead, however, to forms possessing excessive surface energy, and there will often be a tendency for the freshly deposited atoms or ions to re-arrange themselves, possibly by surface diffusion, so that the dendritic form is not developed. As to whether surface-rich or surface-poor forms are observed depends on the relative rates of two processes : ( I ) the deposition of atoms and (2) their re-arrangement. The second change will be influenced by the specific surface energy and the surface mobility of the particles.Certain simple experiments l7 on the production of two-dimensional lead trees by pressing the edge of a vertical zinc strip on a filter paper soaked in a solution containing lead acetate may serve to illustrate the principles involved. Filaments of lead quickly push outwards along the paper, and soon the deposition of lead a t the growing tips is proceeding nearly a centimetre from the place where zinc is being dissolved. The need for this " action a t a distance " becomes clearer on applying a sulphide indicator ; it is found that the lead salt has become almost 1 5 Mott, This Discussion. 1 6 Bunn and Emmett, This Discussion, l7 Evans, Chem. and Ind., 1925, 812.GENERAL DISCUSSION exhausted near the metallic zinc.The lead “ plants,’’ unable to find nutriment at home, push out into the world to obtain it. Of course, given time, lead salts would diffuse towards the zinc. But diffusion under a concentration gradient is generally a slower process than ionic migration under a potential gradient; both are due to the same type of movement, but it is random in the first case and “directed ” in the second. Whether the flow of electric current along the lead filaments has ever been demonstrated may be doubted ; but, in the analogous case of corrosion of zinc by a sodium salt solution and oxygen, the current has been measured l8 and found strong enough to account for the observed corrosion rate. Clearly, dendritic growth can occur without a flow of current; but i t is particularly likely to be met with in the cathodic deposition of a heavy metal where the volume occupied by the metal after deposition is usually much smaller than the volume of solution containing the requisite number of ions, so that, in the absence of stirring, deposition in layers soon becomes impossible if the current, provided from an external battery, exceeds a certain value ; thus “ treeing ” has become one of the electrodepositor’s nightmares.CONCENTRIC GROWTH (IN CIRCLES AND SPHERES) .-Patterson describes the spreading of rust over a pure iron surface as proceeding in a concentric manner, although on steel mossy growths, recalling dendritic forms, are commoner, as shown also by Vernon. O Spherical growth would, presumably, be the normal form of solid accretion in a universe in which surface energy was independent of direction.Even jn our complicated world, it is the normal form of liquid accretion. Matter undergoing a phase change may be regarded as momentarily liquid, and i t is tempting thus to seek an interpretation of the pyramid-growth spreading from face centres, as observed by Bunn and Emmett,16 since this geometrically represents an attempt to attain spherical form. One a t least of those who saw the I.C.I. film was impressed by the resemblance of some of the phenomena to liquid flowing out of a hole. But such resemblances are often misleading, and it is best not to press the analogy too far. The laws of expanding circles and spheres, which are important in connection with surface oxidation and annealing changes, have been developed by a simple method described elsewhere 21 ; in cases where other, more tedious, methods are available, the results are in agreement.When films spread over a surface from pre-existing nuclei (sporadically distributed) which start to operate at zero time (no other nuclei appearing thereafter), the fraction of the surface remaining uncovered after time t will be e-kt’; if nuclei are absent a t the outset, but appear (sporadically in time as well as in space) on the ever-diminishing area available, the fraction will be e-kt’. For expanding spheres, the corresponding expressions will be e-kt’ and e-kf’ respectively. Naturally K has a different meaning (and different dimensions) in the four cases, as shown in Table I, which also includes the final grain-size obtained when the phase change is complete.Where growth has occurred solely from pre-existing nuclei, the grain size must clearly be independent of the crystallization velocity, being the reciprocal of the nucleus number. However, where there are no pre-existing nuclei, it is propor- tional to the appropriate power of the ratio of crystallization velocity to nucleation rate; the numerical coefficient arises out of a gamma function. The power (2/3 and 3/4 in the two cases) deserves notice, since it has been stated that the grain size is given by the first power of that ratio-which is surely impossible, since the expression would then have the wrong dimensions. The distinction between the expressions for spreading in the presence and absence of pre-existing nuclei may come to be helpful in distinguishing between the two cases.In some types of phase change, the points where the change originates are apparently not crystal germs, but seem to be points of atomic disarray where the energy of inception is lower than elsewhere. 18 Agar, quoted by Evans, J . Iron Steel Inst., 1940, 141, 2 2 0 P ; also Noordhof and l9 Patterson, .I. ‘Soc. Chem. Ind., 1930, 49, 206 T . 20 Vernon, Trans. Faraduy SOC., 1924, 19, 887 (Fig. 21). 21 Evans, Tmns. Faraduy SOC., 1945, 41, 365. Evans (unpublished work).GENERAL DISCUSSION 79 ALLOTRIMORPHIC FORMS.-A polycrystalline aggregate formed from a melt in which spherical crystals spread at uniform rate from pre-existing nuclei should consist of grains separated by plane boundaries ; if pre-existing nuclei are absent the boundaries will be curved, whilst, if furthermore the outgrowth is dendritic, the boundaries will be irregular and interlocked-as is often observed. On annealing, the boundaries will tend to straighten-thus diminishing the inter- facial energy of the system. TABLE I SYMBOLS .-v represents the radial velocity of crystallization, w the two-dimensional nucleus number defined by the statement that the chance of finding a nucleus in area element da is wda, Q the two-dimensional nucleation rate defined by the statement that the chance of a nucleus appearing in area element da and time-element 6t is Qdast, whilst w’ and a’ are the corresponding three- dimensional quantities. Conditions Value of k Final Grain Size Two-dimensional : pre-existing nuclei . . 7rwv2 I lw Three-dimensional : pre-existing nuclei . . 47rw’v3/3 I /w’ Two-dimensional : no pre-existing nuclei . . 7rQu2/3 1.137(v/Q)’/3 Three-dimensional : no pre-existing nuclei. . I *I I 7 (71/Q’) ‘ / r ?rQ’v3/3 An early examination 22 of Carpenter and Elam’s studies of boundary migra- tion 23 showed that, out of 53 cases, the apparently concave grain invaded the appavently convex grain in 25 cases, whilst in eight cases the reverse movement was noted ; the other 20 cases were doubtful. The existence of eight apparent exceptions to the straightening rule (“ concave invades convex ’ I ) is not surprising, since the curvature was judged from sections and there was no information about the curvature a t right-angles to the paper. If the two curvatures are uncorrelated, one would expect that the rule would break down once in four times. However, the numbers were felt to be too small for significant conclusions, and plans were made for an extensive research by Cook and the writer which should have provided data for statistical analysis and settled numerous outstanding questions. Preliminary data were published,44 but, owing to Dr. Cook’s departure from Cambridge to take up a post elsewhere, the main research has never been carried out. The writer wishes to thank Dr. Bunn, Dr. Wooster and Dr. Agar for helpful discussion. z 2 Evans, J . Inst. Metals, 1922, 27, 140. 23 Carpenter and Elam, J . Inst. Metals, 1920, 24, 104 (Fig. I t o 21). 24 Cook and Evans, Trans. Amer. Inst. Man. Met. Eng., 1924, 71, 627.

ISSN:0366-9033    

DOI:10.1039/DF9490500066

出版商:RSC    

年代:1949

数据来源: RSC

摘要:

GENERAL DISCUSSION 79 11. NUCLEATION AND NORMAL GROWTH Introductory Paper BY W. J. DUNNING Received 10th March, 1949 The simplest case of crystal nucleation is the separation of two solid solution phases from a single phase on cooling to a point below the critical mixing temperature T,, and it is natural1 to apply Einstein’s theory of Berkeley, Phil Mag., 15112, 254.80 NUCLEATION AND NORMAL GROWTH fluctuations2 to this process. phase, the concentration fluctuates away from the mean. of a fluctuation has the form In small regions of the metastable primary The probability W = eASlk, . (1) and if it is assumed to occur at constant energy, then - AA AS = ~ T ' where A is the Helmholtz free energy, hence W = e-AA/kT . (4 For a concentration fluctuation, AA can be obtained by a Taylor expansion in terms of the mole fractions, it will have the form,* I d2A I d3A A A = - 2 (dN2),, __ A N ' + - 6 (dN3), - A N 3 + .. , . . * (3) The first term of the expansion is zero since A has a minimum value at AN = 0. Near to T,, only the first term of (3) is important, and in Fig. I is shown a plot of A against mole fraction for a temperature just below T,. FIG. I. FIG. 2. The probability of a concentration fluctuation in a supersaturated single phase, represented by point x, is, however, not given immediately by the curvature at this point, because Fig. I refers to macroscopic systems and eqn. (3) involves consideration of microscopic regions. The essential difference is that AA in eqn. (3) must include the surface free energy between the fluctuated region and the bulk phase.3 However, near to T, the surface free energy is probably also proportional to the curvature of Fig.I at x, hence the microscopic AA of a fluctuation will be proportional to the curvature of the macroscopic AlN curve. Small fluctuations of concentration will most probably redisperse again, but larger fluctuations may reach a stage where the curvature becomes zero and they then have an even chance of redispersing or continuing to grow until they become small portions of the new phase. These fluctuations 2 Einstein, Ann. Physik, 1910, 3% 1275. 3 Becker, Ann. Physik, 1938, 32, 128. * These equations are not, of course, precise, they are intended to be illustrative only.W. J. DUNNING 81 would be considered nuclei of the new phase.Under these conditions the probability of nucleation would be given by where a is the activity. Wictorin has carried out measurements of the rate of separation of the two phases from a gold-platinum alloy during annealing at a temperature just below Tc. His results were shown by Borelius to be consistent with the view that the rate of nucleation is governed by the magnitude of the curvature of the macroscopic A / N relation. Eqn. (4) will not be valid at temperatures far below T,, and where a solid precipitates from a liquid phase it is very doubtful if, even in principle, the two branches of the A/N curve (Fig. 2) can be considered as continuous. In Fig. z branch a refers to the solution and branch b to the almost pure solid. The process of nucleation is no longer visualized as the progressive enrich- ment of solute in a small volume.Even if the small volume by fluctuations became almost pure solute, we should still be faced with the problem of the discontinuous change of liquid into solid solute. The fluctuations which lead to the separation of the solid solute are therefore conceived as partaking of crystalline properties from the start. The fluctuations begin as two molecules, some attaching a third, and then a fourth molecule and so on, into lattices of localized molecules, until minute structures are formed of the same crystalline nature as the solid solute. The probability of forma- tion of these crystalline fluctuations is considered to be given by eqn. (2) in which AA is the free energy of formation of this nucleus.On these assumptions this is found to be of the form, where CT is the interfacial free energy of the crystal in contact with the super- saturated solution, and F is the area of the surfaces of the crystal. Again these fluctuations have an overall tendency to redisperse until they reach a critical size (" radius " = yk) given by an equation analogous to the Gibbs-Thomson equation, (5) AA = +aF, . where c, and cI0 are the concentrations of the supersaturated and saturated solutions, M is the molecular weight and 6 the density of the crystal. At this critical size, the tendency to redisperse is equal to the tendency to grow. Above this critical size there is an overall tendency to grow. Fluctuations of this critical size are " nuclei." Using eqn. (z), (5) and (6) the rate of nucleation is given by 16 N ~ M ~ ~ ~ _ - a (7) J = 3 R*S2T* log -- ( c":,), 0 (N is Avogadro's number).This theory of Volmer and Weber was improved by Stranski and Kaischew and later by Becker and Doring.8 These latter considered the detailed mechanism by which the fluctuation grows towards and passes the critical size. Considering the separation of a cubic homopolar crystal, and the crystalline fluctuation at a stage in its growth, the incident molecules are laid down one at a time on the surface. 4 Wictorin, Ann. Physik, 1938, 33, 509. 6 Borelius, Ann. Physik, 1938, 33, 517. 6 Volmer and Weber, Z.$hysik. Chem., 1926, 119, 277. 7 Stranski and Kaischew, Z. flhysik. Chem. B, 1934, 26, 317. * Becker and Doring, Ann. Physik, 1935, 24, 719.82 NUCLEATION AND NORMAL GROWTH This kinetic treatment considers three stages of the growth, the formation of linear chains, the formation of two-dimensional islands on completed surfaces, and the formation of three-dimensional lattices.Becker and Doring assume no restrictions on the possibilities of attachment of the units, but they are able to show that the free energy of the system is smallest when the islands are square and the lattices are cubic. There is a critical cube size and a critical square size corresponding to three- and two-dimensional nuclei respectively. Equations are derived for the linear rate of growth and for the rate of homogeneous nucleation.* A comprehensive experimental check on the theory would require to show that the rates of nucleation and growth have the correct dependence upon the supersaturation, the temperature and the surface and edge free energies, and further that the pre-exponential factor is correct.A partial check has been carried out by Arn~ler.~ He showed that the effect of super- saturation upon the induction period of nucleation (taken as inversely proportional to J ) has a fonn related to that of Fig. 3. Further work with the same object in view is reported in this Discussion by Van Hook and FIG. z. Bruno, and by Bransom, Dunning and Millard. The general impression one gets is that the experiments broadly confirm the theory, though from the latter authors’ results there appears a large discrepancy in the value of the pre-exponential factor. This may have a bearing on the funda- mental nature of the fluctuations.Possibly at some stage in the fomation of the nucleus the fluctuation changes from non-localized to localized molecules. In- formation on this is most likely to be gained by studying highly supersaturated solutions where the critical size is small. Possible techniques for such experiments are considered bv Bransom and Dunning. .. though only resdts for low supersaturatioGs are reported. A paper by Davies and Jones describes experiments in which the nucleation of supersaturated solutions of silver chloride is followed by means of conductivity measurements. They find that the metastable limit of the solubility product varies with the ionic ratio of Ag+ to C1- ions. This observation requires to be accounted for by quantitative theory ; presumably the views of Kmyt and Venvey lo on the structure of silver halide sols would be of interest here.The theory of Becker and Doring was derived for nucleation from the vapour phase. Its application to solutions must be restricted to some extent by diffusion processes. Since many experimental observations on nucleation require the nuclei to grow or be developed to observable or measurable size, its principal effect will be on this growth process. Neumann l1 has con- sidered this matter and it would appear that nucleation studies in unstirred tubes l2 may give results in which diffusion effects would mask any possible correspondence with nucleation theory. Van Hook , in this Discussion, also reports the effect of stirring upon nucleation. Amsler, A c f a Physic.Helv., 1942, 15, 699. 10 Kruyt and Verwey, Symposium 01% Hydrophobic Colloids (Amsterdam, 1938)~ 11 Neumann, in Volmer’s Kinetik der Phasenbildung, p. 209, et seq. 18 Dehlinger and Wertz, Ann. Physik, 1939, 36, 226. * See Bransom, Dunning and Millard, this Discussion, eqn. (15) and (19).W. J. DUNNING 83 lt is, however, in the growth of crystals that diffusion may play its significant role. In the papers on the growth of crystals in Section 11, it is necessary to distinguish broadly between those observations in which the growth of isolated crystals is studied and those in which the average growth of a large number of crystals is studied, and between those in which the solution is stirred and those in which the solution is stagnant. In the latter it appears that diffusion may play a dominant role.Again it must not be forgotten that the theory is a statistical one. In the work of Bransom, Dunning and Millard, average growths in stirred solutions were measured and the interesting point which appears is that , contrary to the theory of Becker and Doring, the linear rate of growth is independent of the size of the crystal. This may mean that either the crystals have a mosaic or lineage structure, each of which requires two- dimensional nucleation and that growth planes are halted at the dis- continuities, or that there is a constant surface density of dislocations, the nature of which is discussed by Frank. In contrast to this work on non-polar crystals, which is broadly in accord- ance with theory, the observations of Bunn, Everett, Berg and Humphreys- Owen are very difficult to reconcile with theory.No connection is found between the rate of growth and the supersaturation at the surface. The growth rate of a single face changes for no apparent reason and similar faces have widely different rates. Further growth starts at the centre of a face and spreads outward, apparently in thick layers (except in the case of non- polar crystals). Bunn is of the opinion that the effects are traceable to the characteristic distribution of the diffusion gradient. This is attributed t o a compromise between a tendency to radial diffusion and a tendency for the crystal to remain polygonal. The result is that the gradients are not quite radial and the surfaces of the crystal are not quite flat. Suggestions have been made that perfect crystals do not grow under mild conditions of supersaturation, and that growth on natural crystals proceeds by a mechanism involving dislocations, Fordham contributes some measurements from which it appears that distorted ammonium nitrate crystals grow more quickly than undistorted. The question of foreign nucleation as opposed to homogeneous nucleation is a subject which properly belongs to Section I11 of this Discussion. It is sufficient to call attention here to the work of Tschennak-Seysenegg,18 who found that nucleation was very specific. For example, sodium acetate trihydrate solutions could not be nucleated with the monohydrate nor with the potassium salt. Incidentally his work on the electrical effects which accompany crystallization is of considerable interest. Chemistry Department, The University, Bristol . l3 Tschermak-Seysenegg, 2. Kvist., 1939, XOI, 2 3 0 .

ISSN:0366-9033    

DOI:10.1039/DF9490500079

出版商:RSC    

年代:1949

数据来源: RSC

摘要:

W. J. DUNNING KINETICS OF CRYSTALLIZATION IN SOLUTION Part I BY S. H. BRANSOM, W. J. DUNNING AND B. MILLARD Received 1st March, 1949 The formation of a new phase from a mother phase can be considered as involving two processes, the formation of three-dimensional nuclei and84 KINETICS OF CRYSTALLIZATION IN SOLUTION the growth of these critical nuclei to macroscopic dimensions. The experi- mental study of the first process particularly is rendered difficult by the fact that in early stages of the formation the two processes take place simultaneously. The nuclei themselves are difficult to observe and count, and it appears necessary to “ develop ” them to an observable size by growth. Such experiments must therefore be designed so that the added complication brought in by the development process can be evaluated and taken into account.Such experimental techniques are, we believe , described in Part I and I1 of our investigations. In this work, the use of salts as the precipitating phase has been avoided, since it was considered that there might be present complications of electro- static origin , arising from diffusion potentials and differential adsorption. In this initial exploration we have therefore restricted ourselves to cyclonite. This material has the advantages of being non-ionic; it is soluble in a number of solvents but is relatively insoluble in water; it is stable and does not decompose in solution ; it crystallizes from many solvents in iso- metric habits which approximate to spheres; it has a high melting point and its crystals are hard and do not easily suffer attrition.Continuous Crystallization.-If a solid (cyclonite) can be precipitated from its solutions in a solvent (concentrated nitric acid) by the addition of another miscible solvent (water), the process of crystallization can be studied in the following manner. The solution and the precipitating liquid are fed continuously and at a steady rate into a vessel. This vessel is provided with an outflow through which, once the vessel is filled, the contents pass out at a steady rate. With efficient stirring, the composition of the outflow is the same as the contents of the vessel. With time the system approaches a steady state in which nuclei are forming at a constant rate, crystals are growing at a constant rate (since the degree of supersaturation is constant) and a constant proportion of the entire contents of the vessel is passing through the outflow in unit time.Let the volume of the contents of the vessel be V , and the rate of efflux v per sec. in same units of volume, this is equal to the sum of the rates of influx assuming that there is no volume change on mixing and crystallization. Let 12, be the number of nuclei which form per unit volume in a short interval of time dt, at time to. At some later time t, some of these nuclei born at time to have been lost through the outflow and those which have remained in the vessel (n) have grown in size under the influence of the steady super- saturation. At the time t, the rate of loss of these crystals is or putting t, - to = 8, the age of the crystals, d6 = dt,, and dn v - __ d e - n y 9 * (3) dY/d6 = f(S) .. - (4) Hence I = j ( S ) 0 , . - ( 5 ) hence = noe-vieV . If the linear rate of growth of the crystals depends only on the supersatura- tion S we can put where Y is an average dimension of the crystal, which for isometric crystals may be considered as the “radius.” Substituting for 0 from (5) into (3) we get * (6) = nee- vrlVf(S).S. H. BRANSOM, W. J. DUNNING AND B. MILLARD 85 For those crystals which were born at a time to, there is therefore a relation (6) between n the number which remain in the vessel and their size. In the stationary state the size distribution of the crystal population in the vessel is stationary and does not change with time. Using relations (6) and (4) we have for this distribution, where n(r)dr is the number of crystals per unit volume of suspension which have linear dimensions between r and r + dr.Thus, if the contents of the vessel are analyzed and the number distribution determined, we have By plotting log n(r) against r we can obtain v/Vf(S) and hence f(S) from the slope and then no from the intercept at r = 0. Experiment a1 The crystallizing vessel A was cylindrical with a hemispherical bottom and had an outflow tube set in the side a t a fairly steep angle. Vessel and outflow were jacketed and water was circulated through the jacket from a thermostat. The solution and diluent flowed from two jacketed Marriotte bottles B, B at constant rates which could be adjusted by altering the levels of the air inlets b, b.The jets of the Marriotte bottles passed into the vessel through the closely fitting lid C, through which also passed a thermometer. This lid carried two semi-parabolic baffles on either side of a propeller-like stirrer. By consideration of such factors as the depth and diameter of the vessel A, the shapes and positions of the propeller and baffles, speed and direction of thrust of the stirrer, it is possible to minimize any tendency of the crystals to sediment and to feed in the reactants so that the local supersaturations do not deviate much from the average throughout the vessel. All these items were mounted on a single vertical rod pivoted at its lower end, and clamped a t its upper end. The rod could be released and the apparatus tipped forward to spill the contents of A into a jacketed filter D beneath which was a weighing bottle E to catch the filtrate. Before beginning the experiment water was circulated round the vessels, the Marriotte bottles adjusted to give the required rates of flow and the stirrer started in the empty vessel.Since the stirrer created a vortex any change in the speed of the stirrer during the experiment is to be avoided, otherwise the effective volume of the vessel changes. Then the influx of the solution and the diluent were started simultaneously and the time to fill the vessel to the point of outflowing is noted. The rate of outflow was noted several times during the course of the experiment to check the constancy of the rates of inflow. The experiment was then allowed to run until most of the contents of B, B had been delivered, hence the amount of fluid passing through the vessel A was about thirty to forty times the volume.To analyze the contents of the vessel the following procedure was adopted. Firstly, a running sample of the outflow was collected in order to determine later the overall composition in terms of the three components ; these figures were checked against the known composition of the reactants and their rates of influx. After noting the temperature, the vertical bar was released to spill the contents of the reaction vessel into the filter D and the crystals rapidly filtered. The filtrate was supersaturated and was later weighed and analyzed and the results used for calculation. The crystals were washed free from mother liquor, and after drying and weighing were ready for particle size analysis.Immediately the sample had been tipped out, the inlet feeds were stopped and the remaining contents of the vessel allowed to stir for 30 min. The supersaturation in the vessel was thereby reduced to saturation, the temperature was noted since there is a slight fall in temperature when there is no heat of mixing being produced and the contents of the vessel again tipped through another jacketed filter and the filtrate Fig. I shows the arrangement of the apparatus.86 KINETICS OF CRYSTALLIZATION IN SOLUTION collected. An analysis of this filtrate gave the solubility in the actual solvent ; this figure could be adjusted to the running temperature by means of the known temperature coefficient of solubility.(I) composition of the liquid phase ; ( 2 ) amount of solid in suspension ; (3) particle size distribution ; (4) degree of supersaturation ; ( 5 ) temperature ; (6) effectjve volume of the vessel ; (7) rate of influx and efflux (v). From these figures a number of cross-checks could be made to ensure that the necessary theoretical conditions were in effect. This adjustment was usually small. The observations recorded were : ( i:; FIG. I. The particle size distribution was obtained by means of a photoelectric sedi- mentometer, a description of which has appeared elsewhere.* By means of this instrument either a(?) or W(Y), the weight of particles which have radii between Y and Y + dr, can be determined. In the present case, W(r) was measured and this is related to n(r) by where d is the density and y is a shape factor, y is equal to 4n/3 if the particles are spheres.W ( Y ) = y Z ( Y ) r3d, * (9) On this basis eqn. (9) becomes * Bransom and Dunning, J . SOG. Chenz. Ind., 1949, 68, 80.S. H. BRANSOM, W. J. DUNNING AND B. MILLARD 87 Fig. 2 a shows this function and Fig. 2 b a typical result obtained in the present experiments. In Fig. 3 the plot of log n(r) obtained from Curve z b is shown; this is derived by using eqn. (9). Eqn. (8) shows that if the theoretical conditions are fulfilled log n(r) should be a straight line when plotted against Y , and it is FIG. 2. FIG. 3. seen that the deviations are small. to (8). shown to be The slope of this curve gives f(S) according The position of the maxima on these W(r) against Y curves is easily which allows a check on the values of f(S) derived from the slope of the log % ( Y ) against Y plot. The intercept of the log w(r) against r line with the ordinate for Y = o gives a value of no.Since, however, the results of the photoelectric sedimentometer are not easily interpretable for crystals smaller than about Y = 10 p, an extra- polation has to be made here, b u t the relative values of %,, and the order of magnitude should be reliable, unless large and varying slopes occur between88 KINETICS OF CRYSTALLIZATION IN SOLUTION - Mother Date of Temp- Liquor Expt. (" C) Comp. % --I--- 54'1 54'4 28.5.42 I 68.8 3.6.42 1 67.8 53'7 5.6.42 ' 66-4 54'7 Average - 8.6.42 1 65.6 9.6.42 , 65.0 - Average 17.6.42 66.0 52.6 2.6.42 1 66.3 53'6 Average 18.6.42 66.0 53'9 __ - Y = 10 p and r = rk the radius of the nucleus.obtained from r,,,. by means of the relation, Another value of n, can be Time of passage (min.) 2.40 58 8-6 -016 2.25 52 7'7 '047 -018 -066 2.60 54 7'6 3'0 52 6.4 2.8 7.9 '039 .030 2.9 57 2'7 56 7'6 '043 84 5'6 6-1 4'0 6.1 70 3'5 -~ 57 6.9 ____ '044 '020 '020 5'7 80 6.0 ~ 78 4'3 -028 2.6 -006 2'9 '010 2'1 -030 14.0 1 120 where y is the yield of crystals per unit of time from the outflow, and no is the number of nuclei formed per unit of time per unit volume. Eqn. (12) serves as a check on the previous value obtained for no. Results Table I gives the results obtained for a series of experiments a t about 67" C ; groups of these experiments were carried out for times of passage ( V / v ) in the regions of 2-5, 6 and 14 min.The third column gives the percentage of nitric acid in the mother liquor of the crystallizing vessel A, and these values were reasonably constant a t 54 yo f I. Definite trends are noticeable between the factors V / v , rmax., the rate of linear growth f(S) and the rate of nucleation. n,/ml. min. 8.4 x 104 9-3 x 104 7-0 x 104 7.2 x 104 6-2 x 104 7.4 x 104 7'3 x 104 1.4 x 104 1-7 x 104 2.6 x 104 1.9 x 104 0-28 x 104 0.20 x 104 0.30 x 104 0.26 x 104 The least satisfactory are the measurements of the supersaturation which exhibit a wide scatter. Nevertheless, comparison of averages in columns 6 and 7 suggests that the rate of linear growth is a linear function of the supersaturation over the limited range studied, i.e., or The trend of no with the supersaturation (eighth and seventh columns) is such that no is a function of a higher power of the supersaturation than the first. An empirical relation would be dv=KSdO .(13) Y = K s ~ . . ( 1 3 4 no = qS3 . - (14) Discussion Whilst the above results are limited in extent and precision they exhibit trends which are worth examination. Volmer has modified the theory of Becker and Doring2 to obtain equations for the linear rate of growth of a Volmer, Kinetik der Phasenbildung (Steinkopff, Dresden and Leipzig, 1939). Becker and Doring, Ann. Physik, 1935, 24 ( 5 ) , 719.S. H. BRANSOM, W. J. DUNNING AND B. MILLARD 89 crystal and the rate of homogeneous nucleation. linear rate of growth of a crystal in cm./sec.) the expression, Volmer gives for g (the a C a10 CIO where pI - pIIw = kT log 2 = kT log - , - (16) CI - G o - k T .- - k T S . . CIO pI, aI, c, are the chemical potential, the activity and the concentration of the solute in the supersaturated solution and pIIm, a,,, cIo the corresponding quantities in the saturated solution, and S is the supersaturation. A’ and A” are the activation energies for the formation of “ one-dimensional and “ two-dimensional ” nuclei on the surface of the crystal ; 6 is the distance between the crystal planes normal to the direction of growth, and x is the length of side of the square two-dimensional nucleus. Since this nucleus is in quasi-equilibrium with the supersaturated solution, an equation analogous to the Gibbs-Thomson equation can be written 9 (17) PS x = P I - PI100 where p is the edge free energy.The factor wlF represents the number of solute molecules which encounter the crystal surface of area F. From this it would appear that the rate of linear growth should be proportional to the radius or to the square of the radius of the crystaL3 If this were so our analysis of the stationary state in the reaction vessel (eqn. (1)-(7)) would not give a Poisson distribution for n(r) and since such a distribution is obtained experimentally, g must be independent of Y . Also wlF may be expected to depend upon the first power of the solute concentration and upon the diffusion coefficient of the solute. With these modifications there results (18) CI g = const. pf(D) e-A’IkT e-A”lkT # - wheref(D) is some function of the diffusion constant D, or A ‘ I kT .p f ( D ) ,-A‘/kT - log cI - __ log g = log ~ T At 340°K and near to it, we will treat the logarithm on the right-hand side as constant since A’/kT may be expected to be of less significance than A”/kT, and we have no data concerning the temperature dependence of D. From 4, A” wMp2N, kT - 2d8R2T2S ’ (20) _ _ - , where w is a shape factor (taken as Z X ) . We now have log g = log (const.) - log c, - 4.78 x 1017 ($3) - From this equation for the rate of linear growth, the edge free energy p may be calculated in two ways, from the dependence upon the supersatura- tion S at constant temperature and from the dependence upon temperature Smoluchowski, 2. physik. Chem., 1918, 92, 129. Volmer, Kinetik der Phasenbildung (Steinkopff, Dresden and Leipzig, 1g3g), p.104 and 183.90 KINETICS OF CRYSTALLIZATION I N SOLUTION at constant supersaturation. The first method is more free from objection since at constant temperature D and the encounter frequency (since the supersaturations are small in these experiments) are constant. The second method is dependent upon the assumption that the temperature coefficients of D and p are constant and that the encounter frequency w,F merely increases proportionally to the increased solubility. FIG. 4. For the first method we have at T = 340' K and cr constant : log g = log dr/dt + const. = const. - 4'14 x 1o12 p" S ' In Fig. 4, log,, dr/dt is plotted against r/S, the values being the averages in Table I. A straight line has been drawn between the three points, and from its slope it is found that p = 7-4 x 10-8 erg/cm.The second method gives for TI = 340 and T , = 349' K with the solu- bilities cI equal to 1-13 and 1-43 g./Ioo g. acid at T , and T , respectively, g2 c12 g1 CKI log - = log - - 4-78 x 1017 Table I1 gives the rate of growth at 349' and S = 0.022 as 8.0 x I O - ~ cm./min. and from Fig. 4 the value at S = 0.022 is interpolated. From these p = 22 x 10-8 erg/cm. TABLE I1S. H. BRANSOM, W. J. DUNNING AND B. MILLARD 91 In a similar manner the free surface energy CT may be calculated from the results in two ways. The theoretical relation derived by Volmer is where A" is again the activation energy for surface nucleation and A"' the activation energy for homogeneous nucleation. Using Volmer's relations we have , - * (21) J ~ re! z 12 e 2XI3kT e- A"/kT e- A'"/kT 1 l k and which now bears a close resemblance to Becker and Doring's equation, except for the factor involving A the heat of solution and a term (pI - -I/S.terms : For our present purpose we will use an equation without the h with we then obtain in the neighbourhood of 340' K, 03 T3S2 ' log J = const. - 3 log S - 2-7 x 105 ___ in which we have included A"/kT in the constant term. The surface free energy CT may be calculated from the dependence at constant temperature of the rate of nucleation no upon the supersaturation S. At 340' the relation becomes c3 log J = log no - const. = const. - 3 log S - 6-9 x I O - ~ - S2 * In Fig. 5, log no + 3 log S has been plotted against z/S2 and a straight line drawn between the points.From the slope we obtain a value of 0 3 = 0.25 (erg/sq. ~ m . ) ~ . Another evaluation may be obtained from the temperature dependence at constant supersaturation. It must be remembered that the encounter frequency w,Z, may be expected to depend upon the diffusion coefficient and upon the square of the solute concentration. We shall for the present purpose assume that the diffusion coefficient is independent of the tem- perature. Thus, log J = log no + const. = const. + 2 log cI - 2-7 x I O ~ ___ The value of no at 349' and S = 0-022 is 5.9 x 104, a value of no at 340' K and the same supersaturation can be interpolated with the aid of Fig. 5, and this is no = 2.84 x 104, Using the values of the solubilities given above we arrive at a value, tJ3 T3S2 * cr3 = 0.27 (erg/sq.~ m . ) ~ . These values give cr = 0.64 erg/sq. cm. Too much stress ought not to be laid upon their concordance, considering the assumptions made, especially ' Volmer, Kineiik der PkasenbiMung (Steinkopff, Dresden and Leipzig, 1g3g), p. 178.92 KINETICS OF CRYSTALLIZATION IN SOLUTION in the latter case and particularly in view of the scatter in the experimental results. The order of magnitude of cr can sometimes be estimated from the heat of solution. The temperature coefficient of the solubility gives a value of k = 4.2 x 10-l~ erg/molecule. Taking the surface area of a molecule as 6a2, this gives cr - 13 erg/sq. cm. This is about 20 times greater than the derived figure but does not make allowance for surface effects such as interphase potentials which will occur only in the presence of the bulk phase, as distinct from the molecularly dispersed phase, and also of effects such as preferential adsorption of one of the solvents.The difference between the estimated value, which is a surface heat content and the required surface free energy due to the surface entropy factor will probably be a minor part of the discrepancy. FIG. 5 If any weight can be given to the degree of concordance between the two values of CJ compared to the wider discrepancy between the two values of 9, an explanation might lie in the relative importance of diffusion in the encounter processes of the two mechanisms. It seems reasonable that di-ffusion will play a greater role in the growth of a large crystal where the large surface demands solute from a linear concentration gradient, than in the formation of a nucleus where the small nucleus is at the centre of a radiating concentration gradient.The following figures give some idea of the orders of magnitude involved. They are calculated for T = 340° K, S = 0-022, p = 7-4 x IO-*, and cs = 0.64. The surface nucleus contains 45 molecules, the three-dimensional nucleus ( n k ) contains 320 molecules, A”/kT = I, A’”/kT = 3.62, and A/kT = 10. The surface nucleus is thus about the size of one side of the three-dimensional nucleus, assuming the latter to be a cube. We included A”/kT with the constants in our calculations of cr. This is justified if we use the value of 7.4 x I O - ~ for p since the variation in this factor is but 10 yo of the variation of A’”/kT with temperature.It is not justified if p is taken as 22 x 10-8. If the latter figure were used to derive a new value for cr from the temperature coefficient of no, a negative value of cr would be obtained.S. H. BRANSOM, W. J. DUNNING AND B. MILLARD 93 The neglect of the term in A/kT in the calculation of the temperature Towards coefficient seems very serious in view of its order of magnitude. the end of his calculation, Volmer 5 makes the following identifications 1 . - (27) 4x2(c2 - e3) = A”’ X (Ez - E 3 ) = A” 3 (E2 - 63) = A We have already seen that there is a discrepancy between E* - E ~ , which is the surface energy per atom and the estimated value of the surface energy as calculated from the heat of solution.An alternative method is to calculate 3(c2 - c3) from our experimental values of A”’ and A”. We then have n k = ( z x ) ~ = 320, whence 4x2 = 46.7. The size of the surface nucleus is 45 molecules, from which x = 3-4. Then and Volmer’s exponent is 2(c2 - c3)/kET which equals 0.15 or 0.5, and so this exponent may be much smaller than the substituted value of 2h/kT which he used. The contribution of this term to the temperature coefficient of no is then only about 1-2 yo of that observed. The most convenient way of comparing the order of magnitude of the experimental and theoretical rates of nucleation is to compare the values of the factor w,Z,. Inserting the values obtained for S = 0.022 and T = 340’ K in Volmer’s equation (23) it is found that log,, w1Z1 = 2’2.For crystallization from a supersaturated vapour, Becker and Doring describe wlZl as the gas kinetic binary collision frequency. By analogy, ZP+Z, in our case should be the binary encounter frequency. Studies of chemical reaction kinetics in solution 6 suggest that the binary collision frequency between solute molecules would be in our experiments of the order 1 0 ~ ~ per sec. This figure probably includes repetitive collisions and it is likely that in nucleation the encounter frequency would be more appropriate. Considerations based on the work of Smoluch~wski,~ Bradley 7 and Olander suggest that the encounter frequency is but one or two powers of ten smaller. The resulting figure for w,Z, of 1 0 ~ ~ or 1 0 ~ ~ is very far from the experimental result. In their calculation Becker and Doring consider only energy terms, and towards the end make the substitutions given in our eqn.(27) and (28), in which total surface energy appears to have been confused with free surface energy. Thus it might appear that entropy terms have been neglected throughout their calculation, and that the large decrease in entropy on crystallization would introduce a factor which would considerably reduce the probability of nucleation. In order to investigate more closely this question, it is convenient to transcribe Becker and Doring’s treatment into Eyring’s nomenclat~re.~ Moelwyn-Hughes, Kinetics of Reactions in Solution (O.U.P.) . Hinshelwood, Kinetics of Chemical Change (O.U.P.). Bradley, J . Chem. Soc., 1935, 1910. Olander, 2. physik.Chem., 1929, 144, 118. Glasstone, Laidler and Eyring, The Theory of Rate Processes (McGraw-Hill, New York and London, 1941).94 KINETICS OF CRYSTALLIZATION IN SOLUTION In thus sketching their treatment, we assume that the partial potential of the solute in the supersaturated solution is given by p I = - k T ( i ) l O g - + ~ , (28) ~ 1 1 = - kT logfs . (29) where x is the mole fraction and fI the partition function of a solute molecule. For the infinite solid, where f' is the partition function of a molecule in the solid. deposition of solute molecules onto the surface of a crystal is given by The rate of where o is a weight factor (number of sites available), Pi is the partition function of the crystal onto which the molecule is depositing and Pi+l is the partition function of the activated state consisting of the crystal and the depositing (or dissolving) molecule.The rate of solution of molecules from the crvstal is given bv At equilibrium between the saturated solution (mole fraction x,,), i.e., ,f,e and fs are measured from the same zero of energy. The system of equations appearing in their theory then takes the form From these, there results The partition function of the transition state can be written as wherefg is that part of the partition function due to the adsorbing molecule. P S m l h + l = Pmlh f' ? We may consider the small crystal to be composed of (m - 2) (h - 2 ) (1 - 2) 2(m - 2) (1 - 2) + 2(1 - 2) (12 - 2 ) + 2(h - a) (m - 2 ) 4(m + 1 + q - 8 interior molecules, each of which has a partition function of f"'mlh ; of surface molecules, each with a partition function of fmJh(s) ; of edge molecules, each with a partition function of fmlh(e) ; and finally of eight comer molecules, each with a partition function fmlh(+ PuttingS.H. BRANSOM, W. J. DUNNING AND B. MILLARD 9s The condition for quasi-equilibrium between the supersaturated solution and the small crystal is given by where 6 takes into account the changes in the partition functions fnzlh when mlh is increased by unity and n = mlh. Inserting this value of - into the equation for J , and taking m = I = h = 2x, we obtain x f1e kT ( ~")'"'i3(f')8121~3(fO) "$8 (39) J z ~ 0, _--- l h Pl Also All the terms except f* and 6 appear in different guise in the Becker-Doring theory, so it is these two terms we must examine.The free energy change for the deposition of the last molecule into the activated adsorbed state on the crystal mlh is given by - (40) -AFz=kTlog--. f h . h e The entropy portion of this change will be due mainly to the loss of the rotational degrees of freedom of the dissolved molecule as it becomes activatedly adsorbed. Since such a complicated molecule as cyclonite must fit precisely into the lattice, this entropy factor may reduce the probability of nucleation by as muchl0 as 10-l~. The factor 6 takes into account the fact that the molecular partition functions in the small crystal vary with the size of the crystal. For example, the oscillation frequency of a molecule in a small crystal will be greater than that in an infinite crystal. The entropy change due to this can be estimated from the decrease in entropy of a crystal under the influence of increased pressure. This is eiven bv However, if likely values of the coefficient of expansion and the pressure change are substituted, the effect proves to be insignificant. It is possible that an examination of the partition functions of small crystals might reveal interesting factors, since the distribution of the normal modes may be expected to depend upon the size to a very marked extent. Another possibility is that the nucleus may not at the critical stage be a small crystal, but may partake of some of the properties approximating to a liquid. A variation of this idea would be a suggestion that the adsorbed monolayer is non-localized. We wish to thank Prof. W. E. Garner, F.R.S., under whose direction this work was carried out, for his interest and encouragement, and also Prof. E. G. Cox, Dr. A. Brewin and Dr. M. Hey for helpful discussion. This paper is published by permission of the Chief Scientist, Ministry of Supply. Department of Physical and Inorganic Chemistry, Thfi University, Bristol. 10 Eyring, ref. 9, p. 19.

ISSN:0366-9033    

DOI:10.1039/DF9490500083

出版商:RSC    

年代:1949

数据来源: RSC

摘要:

KINETICS OF CRYSTALLIZATION Part I1 BY S. H. BRANSOM AND W. J. DUNNING Received 28th February, 1949 In Part I, the continuous method of studying crystallization kinetics was discussed and some results were presented. Although the results obtained are extremely promising, the state of crystallization as a branch of knowledge is so insecure that we consider it necessary to develop a number of different modes of experimental attack. If, then, results are obtained by widely different techniques, any agreement among them can be considered signifi- cant. Criticisms can often be made against experiments the validity of which is difficult to assess. For example, one obvious objection to the continuous process is that the crystals, already present in the stirred vessel, " catalyze " in some way the formation of new nuclei, e.g., small submicroscopic chips may be formed by mutual attrition.Again, in the continuous process it is difficult to exclude foreign nuclei without considerable elaboration. Whilst , if the results are precise, it is possible to exclude such criticisms by a study of the functional dependence on the variables, nevertheless we have considered other methods of investigation, the results of which can be used to check those from the continuous process. Another such technique can be referred to as the " batch process " to distinguish it from the continuous process. In this method a degree of super- saturation is brought about suddenly, giving rise to subsequent nucleation and growth. The initial supersaturation then falls to zero because of the precipitation of the new phase.In the experiments to be described, the initial supersaturation was brought about by the addition of water to a concentrated solution of cyclonite in acetone. The solubility of cyclonite is much less in the aqueous acetone. It is profitable to consider the mathematics of such a precipitation. It will be assumed that the rate of nucleation depends only upon the supersaturation S (where C I - G I 0 s=- C I O c, is the concentration of the supersaturated solution and cIo that of the saturated solution) and not upon the presence or absence of other crystals, nor upon such effects as stirring. In this case we can put : Rate of nucleation = F(S), . (1) where F ( S ) is an unspecified function of the supersaturation.In the same way we will assume that the linear rate of growth of nuclei and crystals is also a function of S alone and does not depend upon the size, rate of stirring, etc. Hence, if Y is a linear dimension, dr/d8 =f(S), . (2) where 8 is the time measured from the initiation of supersaturation. At such a time 8, the distribution of size of the crystals in suspension, can be described by a function n ( r J e ) J * (3) 96S. H. BRANSOM AND W. J. DUNNING 97 which gives the number of crystals present at time 0 that have radii greater than Y. In this notation, the rate of nucleation is Since only nuclei with Y = o are newly formed, the total differential of n(r, 0) with respect to 0 will be zero, i.e., * (5 an(r, 0) an(r, 0) 3r ae ~ + 3 r * s = O or finally the supersaturation S(0) at time 8 is equal to the initial super- saturation s ( ~ ) , less theIamount of solid which has crystallized at time 8, i.e., 0 where d is the density, M the molecular weight, r(0) the radius of the largest crystals, i.e., those which were born when 8 = 0, and o is a shape factor (:= 47r/3 for spherical crystals). We may also write 9 0 S(0) = S(o) - @ I an(o -& t ) { f(S,) dz }'dt , .M (7) 0 t e e i.e., S(0) = S(o) -- $! F(St) . { Jf(S,)dr }'dt , . (74 0 where o < t < O and t < T < e . We now consider applications of these expressions to experiments. If the experiment consists of following the decrease of the supersaturation with time, S(0) is then known, but to derive the functions F(S) andf(S) from this relation would be a laborious task, unless the functions are of very simple f0rm.l Even if the general forms of the functions are assumed to be those given by the Becker-Doring theory,2 it would still be a lengthy task to match up the two sides of the equation since the surface free energies and edge free energies are not in general known.A more promising method would be to consider the relation in neighbourhood of S(o). We can then put as a first approximation on the right-hand side of (74, st = s, = S(0) , * (8) wd and obtain S(0) - S(o) = . F (S(o)) . [ f ( S ( o ) ) l 3 . 8* . If, therefore, S(0) is plotted against 04, the tangent at S(o) will give a value of and the form of this product can be obtained from a series of experiments in which S(o) is varied. Another type of experiment is that in which the increase in the Tyndall scattering of the crystallizing solution is observed.A beam of light is passed through the solution and the intensity of the scattered light measured throughout the course of the precipitation. In the initial stages of the F ~ 0 ) 1 . C ~ W ) 113 Todes, Acta Physicochim., 1940, 13, 617. 2 Becker and Doring, Ann. Physik, 1935, 24, 719. D98 KINETICS OF CRYSTALLIZATION precipitation, the particles will be small enough to scatter light according to Rayleigh's n v2 I == KI, (I + C O S ~ 9) h4 , * (9) where I is the intensity of the scattered light of wavelength h and primary intensity Io, measured at an angle y , the number of particles of volume z1 being n per unit volume. At time 8, assuming spherical particles.Hence when 8 is small. From this relation it may be possible to derive values for for various initial supersaturations. Carried out in conjunction with the previous experiment in which is found both F(S(o) ) and f( S(o) ) should be derivable. Such a technique should give valuable information regarding the initial stages of formation and growth of nuclei. When the crystals grow larger and hence comparable with the wavelength of the light used, deviations from the Rayleigh expression will appear. Observations on the polarization of the scattered light would furnish some information on the shape of the nuclei. Another method of obtaining F(S) andf(S) has been studied by us, in which the final particle size distribution of the precipitated crystals has been used.For 8 = 03 eqn. (6) gives FW(o) 1 [f{S(O) )I6 w ( o ) x") 113 * (12) ,d?-' . ,,3dr, M S(0) - S(c0) = 0 3n(r 00) 3r where is the size distribution of the final precipitate * and ymax. is the size of the largest crystals present. These largest crystals are those born first in the experiment and have therefore been growing for the longest time in the most supersaturated solution. Hence in the final particle size distri- bution, we may immediately identify the largest crystals present as resulting from those nuclei born in the time interval o to de when the supersaturation was S(o). In the same way, all particles in the range re to re + dye of the final distribution were born in the time interval 8 to 8 + de when the supersaturation was SO. Now the number of such crystals in the final distribution is which is equal to the number of nuclei born in the interval 8, 4 + dQ.Hence 3 Rayleigh, Phil. Mag., 1899, 47, 375. La Mer, J . Physic. Chem., 1948, 52, 65. * Note that the symbol hers differs from that used in Part I, there n ( ~ ) was used for 3n(v). a vS. H. BRANSOM AND W. 3. DUNNING 99 or and The right-hand side of the equation is known from the determined particle size distribution, and so the ratio on the left is known. The supersaturation S(0) at which these particles were born can be calcu- lated. From (7a) and (15) we have using (2) we have S(0) = S(O) - odj { r=r* . [ t f(S,) . d - ~ ] ~ . drt, M 0 and if the rate of growth is independent of the size then 4 r J f(S)dz = J dr ; t 'ma. The integration of (18) can be carried out graphically quite readily from the final particle size distribution.The photoelectric sedimentometer gives an(ry 03) against I . In Fig. I we have put arc(ry ____ as n'(r> in closer a plot of - conformity with our notation in Part I. To illustrate the procedure in the graphical integration of eqn. (IS) abscissz rmax., ~ 1 , r2, I,, r8 . . . . . and ar arI00 KINETICS OF CRYSTALLIZATION the corresponding ordinates, nYtlmax., nt1, d2, nt3, nr4 . . . , are marked off. The supersaturations at which these groups were born are then obtained as od l - M S(o) - S - __ . n', . (rmax. - Y , ) ~ , In this manner we can obtain the ratio Fm (= rtb in Fig. I), as a function of So from a single precipitation. In order to separate F(S(8)) fromf( S(0) 1, the following technique appears to be available.From eqn. (7a), f W ) 1 0 0 - (%) = .f{s(e)> . J F(St) * [ f ( s T ) J 0 A A but 0 where A(8) is the total surface area of the precipitate present at time 8 and 'p is a shape factor (= 47r for spheres). Hence If, therefore, during the experiment the changes in the supersaturation and the total surface area of the precipitate can be recorded, thenf(S(8)) can be evaluated. A convenient method of determining A(8) is by means of a photoelectric turbidimeter (similar in operation to our photoelectric sedi- mentometer 6). Experimental A solution of recrystallized cyclonite in acetone-water was prepared and freed from foreign nuclei by developing these slowly to filterable size. More solvent was then added to the filtered solutions and they were then kept at about 35" C for some hours before use.The solution was transferred to the jacketed vessel A (Fig. 2) ; B and C form a second smaller vessel, and C is a flat plate attached to a shaft ; i t carries paddles on its outer edge, which stir the contents of vessel A when the shaft rotates. B is a composite tube consisting of a narrow plate fitting over the shaft of C and a wider portion a t the bottom. The end of this wide portion was ground and polished to fit the surface of plate C, so that when pressed against the plate the two form a liquid-tight vessel. I t was found neces- sary to grease the joint slightly in order to render the small vessel liquid-tight, The tube B was attached to a collar sliding on the shaft of C and driven round with the shaft by a sliding key.This small vessel contained sufficient water to dilute the solution in A to a final concentration of 50 yo by weight. After the two liquids had reached the temperature of the circulating water (about 24.7" C), a trigger was released and the tube B was snapped away from the plate C by means of a spring. This occurred with the stirrer shaft in motion and the diluting water was projected and stirred rapidly into the outer vessel. A small heat rise (about 0.3" C) occurred on mixing and so water from a second thermostat at 25" C was switched in a t the moment of mixing. In this apparatus there was no means of following the decrease of super- saturation, and visu:,.l observation was relied upon to estimate when the precipi- tation had approached completion.The crystals were then filtered by means of Bransom and Dunning, J . SOC. Chem. Ind. 1949, 68, 80.S. H. BRANSOM AND W. J. DUNNING I01 FIG. 2. ===2 - FIG. 3.I02 FIG. 4. 0 FIG. 5.S. H. BRANSOM AND W. J. DUNNING a jacketed filter and the concentration of the solute in the mother liquor was determined and checked against the predetermined solubility. The crystals were weighed and a sample analyzed for particle size distribution. Fig. 3 illustrates the type of size distribution for colonies produced from solutions whose initial supersaturations were less than about 2-5. The number of particles y t ' ( ~ ) with sizes between Y and Y + dr rises to maximum with increasing Y and then drops very steeply to zero. Theoretical considerations suggest that there should be a sharp cusp a t the cut-off at rmax., the rounding of the curve at this point which is found experimentally is no doubt due to limitations in the experimental technique.The cut-off in the region of small radii is probably due to the loss of the smaller crystals through the filter. These size distributions were treated as described above in order to obtain F ( S ( e ) ) as a function of So, and it was further assumed that f{W) } f w 4 ) = w% where R is a constant. were obtained. Fig. 4 gives the plots of these values as derived from a series of experiments in which the initial supersaturations were varied. It is seen that the curves superimpose on each other. This implies that the rate of nucleation depends only upon the supersaturation and not upon the presence of crystals.Hence there is no evidence for auto- or secondary-nuclea- tioqs nor does the stirring cause attrition with the formation of small centres for crystallization. Furthermore, it is seen that the points for initial nucleations also lie on the curve. Since the initial supersaturations are known from the quantities of the solvents and solutions which were used, the integrations were dispensed with and the particle size distributions merely used to obtain the number of particles with the maximum radius. In this way the rates of initial nucleation in the initial supersaturations were determined for a number of solu- tions. The results are shown in Fig. 5 , where F(S(o)}/k is plotted against S(o). Since R , or better f(S(o)}, is not known, the ordinates are relative in magnitude. However, it is seen that F(S)/k is a steeply rising function of the supersaturation. It has not been considered profitable to examine these relationships in greater detail, e.g., in relation to the Becker-Doring theory, since the results are pre- liminary and serve mainly to illustrate an experimental technique. We wish to express our thanks to Prof. W. E. Garner, Prof. E. G. Cox, Dr. M. Hey and Dr. B. Touschek for the interest they have shown in the work. The paper is published by permission of the Chief Scientist, Ministry In this way, by multiplying by S(O), values of ~ Few) 1 k of Supply. Department of Chemistry, Bristol University . Altberg and Lavrow, Acta Physicochim., 1940, 13, 725.

ISSN:0366-9033    

DOI:10.1039/DF9490500096

出版商:RSC    

年代:1949

数据来源: RSC

摘要:

S. H. BRANSOM AND W. J. DUNNING THE PRECIPITATION OF SILVER CHLORIDE FROM AQUEOUS SOLUTIONS Part I BY C. W. DAVIES AND A. L. JONES Received 26th January, 1949 When a substance separates spontaneously from its supersaturated solution the kinetics of the process can be expected to be most complex, at any rate in the early stages ; for the growth of existing nuclei and the formation of new ones may be proceeding concurrently, and the number, size and size distribution of the crystal nuclei may be changing rapidly with104 PRECIPITATION OF SILVER CHLORIDE time. The experimental study is further complicated by the difficulty of exactly reproducing the conditions under which spontaneous nuclei formation is induced, either by gradual temperature variation or by the mixing of solutions; and a foreign substance present in traces, solid or dissolved, may be expected to exert an influence out of proportion to its concentration. In view of all this it is not surprising that the published measurements of crystallization rates have been difficult to interpret, and that such theories as have been advanced still lack convincing experimental support.In our work we have tried to simplify the problem by first studying the behaviour of silver chloride under such conditions of concentration that fresh nuclei are not being formed. The measurements reported here help to define these conditions. Experiment a1 Precipitation was followed by measuring the conductivity of the solution. The amount of AgCl precipitated a t any moment was calculated from the equation : where n is the number of g.-moles precipitated, Ax the corresponding change in the specific conductivity of the solution, z1 the volume of solution in ml.and A the sum of the ionic conductances of the silver and chloride ions. The latter quantity could be regarded as a constant, within experimental error, in the solutions investigated which were all within the range of ionic strength I - 3 x I O - ~ ; A = 138.26 -991.7 (3 x I O - ~ ) ' ! ~ = 137.76, where 91.7 is the Onsager slope. The concentration of KCI or AgNO,, when required, could be calculated in a similar way from theconductivities, or vice versa, using the mobility values 1: 73-52 for K+, and 71-44 for NO,-. The measuring bridge was of the type previously described,2 and, with the low conductivities studied in this work, enabled changes of the order of 0.01 yo to be observed. The solutions were contained in silica or borosilicate cells of the Hartley and Barrett type with greyed platinum electrodes.One was fitted with an efficient Pyrex glass stirrer which could rotate below the electrodes ; all experiments involving seed crystals were carried out in this cell, the normal stirring rate being 250-300 rev./min. The cells were immersed in an oil thermo- stat controlled a t 25O C f: O-OIO, and the room was thermostated 25O f IO C , During runs the cells were either sealed, or a gentle stream of pure air or nitrogen was passed over the solutions ; under these conditions blank experiments showed that variations in the C0,-content of the water-probably the largest single source of error in the measurements--could be controlled, and constant conduc- tivities maintained over many hours.The cells were calibrated by measuring the conductivities of very dilute KC1 solutions a t 25O C, and applying the inter- polation formula 2: A = 149.92 -93.85 C'lr + 50C. The ' cell constants ' so calculated varied from the mean value by not more than 0.04 yo. The water used for the experiments was obtained from a modified Bourdillon still ; its specific conductivity varied between 0-2 and 0.6 x I O - ~ ohm-1. KCZ and AgNO, were both A.R. reagents. After every experiment the cell used was washed with ammonia to remove all traces of solid silver chloride. Seed crystals of AgCl were prepared by crystallization from boiling, saturated solutions.Freshly precipitated AgCl was washed six times by decantation, portions of the precipitate were boiled in 2-1. volumes of distilled water and, after immediate filtration, the solution was allowed to cool very slowly in the dark. The crystals thus formed were washed with conductivity water, and suspensions made up and aged at 2 5 O in darkness for a t least a fortnight. The i t was given the value MacInnes, Shedlovsky and Longsworth, J . Amev. Chem. SOC., 1932, 54, 2758. Davies, J . Chem. SOC., 1937. 432.C. W. DAVIES AND A. L. JONES MI. seed added seed concentrations were determined gravimetrically with a I yo accuracy and were arranged to be of the order of I mg. AgCl per ml. suspension. Microscopic examination showed that the seed crystals were rectangular plates or cubes having an average size of 5-10 p, and occurred singly or in very small clusters. Dura- Final concn.x 105 tion of expt. (hr.) [&'I I [Cl'l Results The Solubility Product at 25" C.-In these determinations the equilibrium was approached from both sides. The precipitation experiments were arranged so that precipitation should occur only on the aged seed crystals. A stable supersaturated solution (see later) of AgCl was prepared in the cell, and C0,-free air passed through until the conductivity became constant. A known volume of a homogeneous seed suspension was then added, and the resulting decrease in conductivity followed to the equilibrium value, from which the solubility product of AgCl was calculated. The results are in Table I.1.613 2.161 TABLE I 1.623 1.097 No. of Expt. ~ - - I0 48 I4 48 5 40 7 46 45 76 78 87 86 1.873 1'844 1.849 1.848 Initial concn. x 105 [Ag-l 1 [Cll] 1-84 1-85 1.81 1.81 1-80 In a second series of experiments, larger quantities of seed were added to water of known conductivity in the cell, and the process of solution followed to apparent equilibrium. The solubility of AgCl was calculated from the final conductivity. A correction for traces of impurity in the seed suspension was obtained by adding further amounts of suspension after equilibrium had been attained, and noting any resulting change in the conductivity. The correction never exceeded 0.7 y/o. The results are given in Table 11. TABLE I1 s x 1010 The suspension used in Expt. 53 and 54 was four times, and that of Expt.80 and 81 ten times, more concentrated than that used in the experiments of Table I. The two methods are in good agreement, and give 1-82 x 10-1~ & 0.02 for the concentration solubility product, or 1-35 x 10-5 & 0.01 g.-mole/l. for the solubility of silver chloride a t 25" C. This value is confirmed by a third method to be described in Part 11. Introducing activity coefficients calculated from the Debye-Huckel limiting equation the true (activity) solubility product becomes 1-81 X IO-lo. Landolt-Bornstein quotes nine previous determinations of the solubility ranging from 1-20 to 1-47 x 10-5 g.-mole/l. ; the average of these values is also 1-35 x 10-5. 3 Landolt-Bornstein, Physik.-Chem. Tabellen (Springer, Berlin 5te. Aufl., 1923). 634 ; Erg.11, 343 ; Erg. 111, 483. D*106 PRECIPITATION OF SILVER CHLORIDE Precipitations in the absence of seed .-Precipitation may be initiated by bringing an AgNO, solution of the approximate concentration 1.35 x I O - ~ to temperature equilibrium in the cell, and then adding from a weight burette a sufficient quantity of a more concentrated KC1 solution to exceed the solubility product. Three such experiments are illustrated in Fig. I, Curve I refers to a run in which 2 ml. 0.004 N KCl solution were added, curve 2 to one in which 10 ml. 0.0008 N solution were used and curve 3 to one in which 20 ml. 0.0004 N solution were employed ; in this last case the KC1 was added in portions over a con- siderable period of time, the final addition of I ml. producing precipitation.The three curves are strikingly different, and i t is clear that the course of the precipitation is governed mainly by the local concentrations of the ions a t the moment of mixing. The number of nuclei available for the subsequent separation of AgCl is greatest for the experiment of curve I and least for curve 3. It may be added that runs carried out in this way are not reproducible, and the final conductivities reached after many hours always correspond to solubility products greater than 1-82 x 10-10, indicating that the resulting crystals are small enough to show an enhanced solubility. Curve I gave a final concentration product of 2-02 x 1 0 - 1 ~ and curve z a value of 1-93 x 10-10, in agreement with the view that more nuclei were available in the faster run which resulted from the higher local concentration on mixing. The run corresponding to curve 3 was not complete after 40 hr.FIG. I .-Precipitations from unseeded solutions. To avoid these local concentration effects it is necessary to mix solutions of approximately equal concentrations, and a number of experiments were made in this way. The cell was half-filled with a solution of one of the reagents, and when temperature equilibrium had been established an equal volume of the other reagent, preheated to 25O, was introduced and the cell gently shaken. A few runs of short duration were inconclusive, but Table I11 contains the results of all experiments by this technique which were followed for a t least three hours. In Expt. 28-1 the total fall in conductivity over 12 hr.corresponded to a change of no more than z mm. in the bridge null-point, and was within the possible experimental error. With this exception all the solutions with a concentration product of 3-14 x 10-10 or less showed no precipitation after periods extending up to 18 hr., and this was true whether the solutions were mechanically stirred, occasionally shaken or agitated by the passage of a rapid stream of nitrogen. The experiments suggest therefore that such solutions, in which the solubilityC. W. DAVIES AND A. L. JONES 107 so. of Expt. 36 24-3 2 8-2 24-B 2 8-1 2 8-3 2 4-U 2 4-6 24-4 31 35 2 4-2 2 4 4 TABLE I11 Initial concn. x lo5 1-88 1-71 1'755 . 1.76 1.76 1-78 1-78 1-79 1-80 1-88 1-86 1-96 2-09 [C] TI 1-48 1'71 1'755 1-76 1.76 1-76 1-78 1-77 1-80 1-88 1-89 1'94 2-09 [&I [Cl'l x I 0 1 0 2-80 2.92 3-08 3-10 3-10 3-14 3-16 3'17 3'2 4 3'53 3'53 3-80 4'3 7 -~ - Duration ~ - 4x x 1061 (hr.) i hr.I5 7 9 I4 18 8 I9 3 40 3 I2 7* 2 0 ' 0 l o i o i 00'003 j o.oo8 0.009 I 0.013 1 0.081 0.080 0'1 I 1 0.26 product is exceeded by more than 50 yo, will remain almost indefinitely without any crystal formation. The critical concentration product above which nuclei development occurs spontaneously is clearly very near 3.14 x 10-l~. To help in fixing it more closely we have used the data for the very slow precipitations which occur when the concentration product just exceeds this value. The last column of Table I11 gives the average rate of fall in conductivity over the first hour or two after mixing. These values are not very accurate, as the smallest of them is not much greater than the possible experimental error, whilst for the fastest runs the values begin to depend on the time interval chosen.Nevertheless, when plotted, as shown in the left-hand curve of Fig. 2 , they give a reasonably good straight line which fixes the critical concentration product a t 3-14 X 10-l~. FIG. 2 .-Ratus of slow precipitations against ionic concentration product. The effect of ionic ratio on the critical supersaturation.-Further series of experiments similar to those described in the last section have been carried out with initial [Ag']/[Cl'] ratios of 0-25, 0.5, 2 and 4. These show quite definitely that a critical supersaturation limit exists in each case, but that its value depends108 PRECIPITATION OF SILVER CHLORIDE on the ionic ratio.The results for a Z/I ratio are the same (within experimental error) whichever ion is in excess, and the same is true for a 4/1 ratio. Mixtures which failed to show any precipitation are listed briefly in Table IV. TABLE I V I Concn. ratio 1 product EAg-]/[Cl'] , x 1ol0 3-15 3-29 3'37 3'42 3-38 I 3-41 I 3'52 3'56 ~ 9 5 14 5 It will be seen that the critical supersaturation is increased by a disparity in the concentrations of silver and chloride ions. To fix the critical values for each concentration ratio more closely, further slow precipitations were carried out, and the results of these are shown in Fig. 2. They lead to the following values- Ionic ratio .. .. . . 1/r Critical concn. product x rolO.. 3.14 112 = /4 3'44 3'59 FIG. 3.-Rate of precipitation in g.-equiv. x 1o*/1. min. plotted against amount precipitated in g.-equiv. x 106/1. Finally, some further evidence concerning the effect of ionic ratio has been obtained in an entirely different way. A number of moderately slow precipitation runs, of the type illustrated in curve 3 of Fig. I, have been carried out and from the conductivity-time plots curves have been constructed showing the rate of precipitation plotted against the amount of silver chloride precipitated. TWO of these are illustrated in Fig. 3 ; they show that the precipitation accelerates to a maximum, and that shortly after this the rate curve changes, at a fairly well-defined point, to a steady linear (or almost linear) decrease. We were inclined to interpret these turning-points as representing the stage in the precipita- tion a t which fresh nuclei cease to be formed.If this i s so, the concentration product at the turning-point may be identified with the critical supersaturation for the ionic ratio holding a t the turning-point. Critical values based on thisC. W. DAVIES AND A. L. JONES 109 hypothesis are compared in Fig. 4 with the directly determined values, and the agreement is excellent. It should be added that in the runs to which this method has been applied the turning-point is not reached until more than an hour after mixing the reagents. More rapid runs have shown fairly abrupt turning-points in their rate curves, but these have not corresponded with the directly determined critical supersatura tions.Ao' in excess Discussion The behaviour a t 25' C of unseeded supersaturated solutions of silver chloride may be summarized as follows. I. Spontaneous precipitation will not occur unless the product of the ionic concentrations is almost double the normal solubility product. 2. The value of this critical supersaturation is markedly dependent on the ionic ratio, but not on which ion is in excess. The data shown in Fig. 4 may be represented by the equation : Sc x I O ~ * = 3-74 - 0.60 n, where Sc is the critical concentration product and n is the ionic ratio expressed as a fraction less than one. 3. If the critical concentration product is just exceeded precipitation always occurs, although it may be so slow as to be perceptible only after several hours.At slightly higher concentrations a characteristic rate curve is given from which, again, the critical supersaturation can be calculated, If the concentration is further increased the kinetics of precipitation become more complex, and when the initial concentrations are about ten times the normal solubility , the precipitation is almost instantaneous. As is well known, Ostwald believed that if the concentration of a solution be gradually increased, the region of stable unsaturated solution is followed, after the normal solubility curve, by a metastable region in which crystal- lization will not occur without suitable inoculation; and that this again is succeeded by a region of labile solutions which crystallize spontaneously. 40stwald, 2. physik.Chew., 1897, 22, 289.I10 PRECIPITATION OF SILVER CHLORIDE This view was supported by the work of M i e r ~ , ~ Hartley,6 Mouat Jones 7 and others, who plotted for many salts the course of the ' supersolubility curve' which separates the metastable from the labile region. I t was criticized by de Coppet,8 whose results were less regular, and who thought that sporadic crystallization was liable to occur, perhaps after long periods of time, in any supersaturated solution; and by Young,9 who states that crystallization from the metastable region can always be induced by violent mechanical shock. In view of de Coppet and Young's criticisms, later writers l o have tended to regard as unreal any rigid distinction between " labile " solutions (in which they consider that crystallization is rapid and easy) and " metastable " solutions (in which it is slow or more difficult), and this view has been quoted in a recent review.ll It will be evident that our results support the earlier belief that metastable solutions can exist up to a definite limit, and this limit can be fixed with considerable accuracy and varies in a regular way with the composition (ionic ratio) of the solution : this applies to solutions under ordinary conditions ; the abnormal conditions studied by Young introduce fresh considerations.Our results also differ from those of de Coppet in that silver chloride invariably precipitates even in unstirred solutions as soon as the supersolubility is exceeded. The Gibbs-Thomson relation may be applied to the solubility of silver chloride particles in the form : where y is the interfacial tension, V the molecular volume of the solid salt, (Ag-1, [Cl'], the concentration product of a solution which is in equilibrium with crystals of (assumed uniform) average linear dimension I,, and a is a numerical factor depending on the shape of the particles ; when I, becomes large, [Ag.], [Cl'], becomes the normal solubility product.This equation cannot be used without a knowledge of the interfacial tension, and moreover it involves the assumption that the interfacial tension is independent of particle size ; nevertheless it is qualitatively valid. It was used by Hartley and Thomas to account for the metastable region. They assumed that crystal nuclei might not attain a size at which they could act as centres of further growth until the supersolubility curve was reached.This idea was extended by later workers,' so as to accommodate de Coppet's views, by supposing that chance encounters in the metastable range may occa- sionally give rise to a particle large enough to initiate crystallization. To serve as a useful basis for discussion these conceptions must be stated with greater precision. If the concentration of a seed-free solution were uniformly increased through the normal solubility value, the rate of growth of any nuclei, however arising, would be increasingly favoured as compared with the rate of solution, until nuclei of a size satisfying the Gibbs-Thomson equation would be eventually produced. The corresponding concentration product would represent the critical supersaturation.Up to this point the nuclei would be unstable, the rate of loss by solution far exceeding, at first, the rate of molecular deposition, and we therefore think that the 6 Miers, Phil. Trans., 1904, 202, 459. Miers and Isaac, Proc. Roy. SOC. A , 1907, 79, 6 Hartley and Thomas, J . Chewz. SOC., 1906,89, 1013 ; Hartley, Jones and Hutchinson, 7 Jones, J . Chem. SOC., 1908, 93, 1739; 1909, 95, 1672. 8 de Coppet, Ann. Chzm. Phys., 1907, 10, 457. 9 Young, J . Amer. Chem. SOC., 1911, 33, 148, 1375 ; 1913, 35, 1oG7. 10 Ting and McCabe, Ind. Eng. Chew., 1934, 26, 1201. l1 Wells. Ann. Reports, 1946, 43, 85. 322 ; 1910, 82, 184 ; J . Chem. SOC., 1906, 86, 413 ; 1908, 93, 927. ibid., 1908, 93, 825.C. W. DAVIES AND A. L. JONES 111 main mechanism of growth is by the successive coalescence of smaller particles; it is only above the supersaturation point that this mechanism may give way to growth by molecular accretion.If this is correct, the number of nuclei attaining a given size will vary very rapidly with changes of concentration, and the critical supersaturation might be identified with a narrow range of concentration in which stable nuclei arise in significant numbers. A consequence of this view would be that stable nuclei can exist in small numbers even in the metastable region; but when we remember that a reduction of concentration will not only result in a very rapid drop in the number of nuclei of given size, but will also lead to a rapid increase in the minimum size of a stable nucleus, it is clear that the chance of detecting crystallization at a point well within the metastable region is vanishingly small.This viewpoint is reconcilable with the results of Hartley, Jones et al. Our results have shown that if precipitation does not actually cease at the critical supersaturation, it must at least become so slow that no change would be detected over a period of days. And although the Oxford workers made many hundreds of experiments without once observing crystallization below their supersolubility curve, their method of cooling would not have enabled them to detect a very slow process. It is possible that the theory also explains our own results : for we have no theoretical basis for the linear extrapolations of Fig. 2 ; and although we have shown that for all practical purposes the critical supersaturation is sharply defined, the experimental distinction can only be between solutions that do or do not show a perceptible change in a reasonable time. ADDENDUM (13th April, 1949) : Further experiments in seeded solutions have now shown that the rate of crystal growth in slightly supersaturated solutions follows the equation : v = kA2, where v is the velocity of crystallization, k a constant, and A is the quantity of silver and chloride ions to be deposited before equilibrium is attained. A result of this behaviour is that, for equal ionic concentration products, the rate of crystal growth will be the smaller the greater is the disparity between the individual ionic concentrations ; the effect is illustrated by the following results for the initial rate of deposition from solutions in which the ionic concentration products are roughly equal : - 1 1 [Ag'I/[Cl'l * ' .. I 2 4 [Ag'] [Cl'] x lo10 . . 2.605 2.585 2.560 2.588 2.560 10s x z, . . . . 1.84 1-54 1.48 1-15 1-06 10-3 x vlA2 . . 2-32 2-54 2-58 2.71 2-60 This provides an explanation of the ionic ratio effect illustrated in Fig. 4 (and it is no longer necessary to assume that coalescence plays a major part in nucleus formation) ; a nucleus of stable size grows more slowly if the ionic ratio is not unity, and the probability of a nucleus growing to stable size within a given limit of time should be reduced in a similar way. We wish to thank the D.S.I.R. for a grant to A. I,. J. Edward Davies Chemical Laboratories, A berystwyth, Wales.

ISSN:0366-9033    

DOI:10.1039/DF9490500103

出版商:RSC    

年代:1949

数据来源: RSC

摘要:

NUCLEATION AND GROWTH IN SUCROSE SOLUTIONS BY ANDREW VAN HOOK AND ARTHUR J. BRUNO Received 3rd February, 1949 Schweizer has pointed out the difficulty of preparing, by ordinary procedures, supersaturated solutions of highly soluble materials which do not exhibit a tendency to nucleate spontaneously. He was able to prepare stable, supersaturated solutions of sucrose and other substances, which were 1.8-fold supersaturated on a sugar to water basis, and which did not crystallize over periods of several months. On the other hand, Waterman and Gentil found that all oversaturated solutions of sucrose crystallized, given sufficient time. Both behaviours find support in the voluminous literature on the nucleation of solutions and melts, e.g., Cassel and Landt,3 K~charenko,~ Meyer and Pfaff,5 Dorsey,6 Volmer,’ Stranski,8 etc., on the basis of the heterogeneous or thermodynamic theories ; Richards, Tammann,lO Van Ginnekin and Smit,ll Fouquet,12 etc., on the basis of the homogeneous theory.It is the purpose of this paper to review this situation for sucrose solutions, from the theoretical and practical viewpoints. Such matter has been considered previously by Cassel and Landt,3 Naveau l3 and Ca~e1le.l~ Preparation of Stable Supersaturated Sucrose Solutions.-Both Schweizer’s and Waterman and Gentil’s experiences were confirmed, using their respective techniques. However, in the latter procedure, which involves dissolution in sealed tubes, it was observed to be possible to prepare slightly oversaturated syrups which have not crystallized over several months, if fine, alcohol-precipitated material was used and/or complete solution was assured by prolonged rotation at temperatures at least zoo above the saturation point.If these precautions are not observed, or if the supersaturation is too high (> 144, crystallization inevitably occurs. It was likewise found possible to duplicate Schweizer’s experience with solutions prepared by means of quick, active boiling, followed by curing (after sealing, or covering with a thick layer of Nujol oil) for at least 20 min. and zoo above the saturation point of the final solution. Presumably the potential nuclei, otherwise preserved, are deactivated by this treatment. One is limited to prepare, at most, approximately 80 yo solutions by either technique ; since beyond this concentration either degradation is unavoid- 1 Schweizer, Rec.trav. cham., 1933, 52, 678 ; I n t . Sugar J . , 1933, 35, 385. 2 Waterman and Gentil, Chem. Weekblad, 1926, 23, 345. 3 Cassel and Landt, 2. dtsh. Zuzker-Ind., 1927, 77, 483. 4 Kucharenko, Planter Sugar Mfg., 1928, 75. Meyer and Pfaff, 2. anorg. Chem., 217, 257 ; 222, 382 : 224, 305. 6Dorsey, Trans. Amer. Phil. Soc., 1948, 38, 248. Volmer, Kinetih der Phasenbildung (Steinkopff, Dresden, 1939). * Stranslri, Physik. Z., 36, 393 ; Ann. Physik, 23, 330. Richards, J . Amer. Chem. SOC., 1936, 38, 2243. 10 Tammann, Kristallisieren und Schelmzen (Barth, Leipzig, 1903) ; States of Aggrega- tion (Van Nostrand. N.Y.. 1925). 11 Van Ginnekin and Smit, Chem. Weekblad, 1919, 16, 1210. 12 Fouquet, Compt.rend., 1910, 150, 280. 13 Naveau, Sucre Belge, 1943, 62, 310, 336. l4 Capelle, Sucre Belge, 1943, 62, 335. I I2ANDREW VAN HOOK AND ARTHUR J. BRUNO 113 able l5 or crystallization sets in.16 This limit is equivalent to a super- saturation of 2.0 at the usual observation temperature of zs0 C, but may be increased slightly to about 2.4 by cooling to - I o O C . This temperature is a lower limit set by the ice-sucrose eutectic point. Sugar/ Water Sugar/ Water at Saturation Supersaturation, 0 = ___ FIG. ti time of appearance of crystals in sucrose syrups prepared by vacuum evapora- tion or active boiling, and curing at 20° C above saturation. Points are individual samples at 25' C unless designated as the average of several samples, or other temperatures. I signifies partly degraded solutions. The times at which the beginning of crystallization was observed in solu- tions prepared in this way are presented in Fig.I. The usual observation temperature was 25' C, others being properly designated on the Figure. The tubes were rotated slowly and the usual sample was about 10 g. solution. Inversion, by test, and/or degradation, by colour, was apparent in solutions above 0 = 2.0. None the less, observations were made in this higher range, and while some results suggest a monotonous extension of lS Montgomery and Wiggins, J . SOC. Chenz. Ind., 1947, 66, 31. l6 Stare, Chern. Zlb., 1940, 2, 2826 ; A.C.S. Abstr., 36, 6033.114 NUCLEATION AND GROWTH IN SUCROSE SOLUTIONS the curve, others indicate a radical change in its nature.It is considered, however, that this change is caused by the impurities present ; for addition of invert sugar, caramel or degraded syrups to pure syrups at lower con- centrations, greatly prolongs the time required for crystallization. The data suggest the stability of solutions less than about 1.6 super- saturated,17 and the very rapid onset of nucleation above this concentration. The beginning of crystallization in syrups, prepared in the ordinary manner may be represented empirically by equations of the form : (0 - a) t = const., where 0 is the supersaturation, t the time and a a constant: a was evaluated as 1-0, 1.2 and 1.05 in three cases surveyed, and 1-37 from some of Waterman and Gentil's2 data. These are tantamount, of course, to Ostwald's metastable limit.Effect of Stirring.-Increased rate of rotation of the tukes had no appreciable effect on the observed nucleation times. Neither did glass propeller stirring, under oil, up to 300 rev./min. and below 0 = 1.4. Above this concentration, however, the nucleation times were greatly reduced the higher the concentration and faster the stirring. For instance, at Q = 1-4, stirring at 300 rev./min. for two days did not especially encourage crystallization. At IOO rev./min. a 1.5 supersaturated syrup crystallized in 4$ hr., whereas without stirring or with gentle rotation it is normally stable for weeks. At 0 = 2.0, where unstirred solutions take about a day to develop a visible crystal, a cloud shows up within a few hours at IOO rev./min. and in about an hour at 300 rev./min.Any accidental contact of the stirrer with the sides of the container, or with added glass beads, induces crystallization very promptly, even at low supersaturations. The foreign, suspended material of ordinary refined sugar seems to have no appreciable effect upon the nucleation time, provided the curing treatment is sufficient. These irregular results with stirring suggest the influence of viscosity ; which factor, therefore, was investigated by means of the temperature coefficient of reaction. Three tubes in a set, at a constant supersaturation of 2.0 with respect to oo, 2-5' and 40' C, were rotated slowly. The times of nucleation noted were remarkably uniform. If the rate of nucleation is taken to be inversely proportional to the time, and the energy of activation assumed constant between each pair of temperatures, the following activation energies are computed.TABLE I TIME OF NUCLEATION AND ENERGY OF ACTIVATION, AT 0 = 2.0 Time I EAct. 1 EAct. for growth l* (kcal./mole) Temp. O C 1 (kcal./mole) I _- l- I --I 0 25 40 5'3 10.6 24'4 I 1.7 Effects of Surface- active Agents .-The addition of surface-ac tive materials in minute amounts had no significant effect, contrary to expectations from discussion in the literat~re.~ l3 l4 The action of Aerosol OT (octyl sodium 17 This is equivalent to a supercooling of about 50°, which is somewhat larger than those reported for many melts and solutions ; Van Hook, A n n u a l Tables of Physical Constants (Princeton, N. J.) (in progress). 18 Van Hook, I n d . Eng. Ckem., 1945, 37, 782.ANDREW VAN HOOK AND ARTHUR J.BRUNO 11.5 sulphosuccinate), which is summarized in Table 11, is typical of the many different types which were studied. The absence of any marked effect confirms our earlier experience l9 that these agents alter neither the nucleation nor growth kinetics of sucrose solutions. However, when nucleation does occur in their presence it is much more prolific than otherwise.20 TABLE I1 EFFECT OF *AEROSOL OT ON THE TIME O F NUCLEATION O F SUCROSE SOLUTIONS Supersat. ' 1 - p Appear- (2j") ' I . , ance Discussion The times reported represent the sum of the time required to establish at least one stable nucleus, and the time for this embryo to grow to visible size. There is likewise the disturbance involved in the transfer from the curing temperature to that of the bath.The growth time is undoubtedly short at all but very small supersaturations ; while the transfer factor is common to all observations and will only alter the position of the curve and not its nature.2122 This shift cannot be appreciable in the present instance, since essentially the same results are obtained under various treatments. The performance reported here is definitely contrary to the homogeneous theory of nucleation as espoused by Tammann and his school.l* Any straightforward heterogeneous theoryJ6 in the sense of foreign n ~ c l e i , ~ 5 likewise seems inapplicable ; since variable curing (provided this is at least zoo above the saturation point, yet not so severe as to hydrolyze or degrade the sugar) has no effect upon the observations.It seems quite clear that those concentration fluctuations which form at least critical-size nuclei are the origin of the crystallization observed in these experiments. The Volmer theory' for condensed systems is adequate to explain the results. and extended by the absolute reaction rate theory,24 suggests that the rate of nucleation is This theory, as modified by the influence of where AF* is the free energy of activation involved in forming the nucleus, l9 Bruno, M.S. Thesis (Holy Cross College, 1947). 2o Van Hook, Ring Surface Tensions (in preparation). Highly concentrated sucrose solutions apparently salt out even traces of most of the surface-active materials investigated. 21 Othmer, 2. aizorg. Chem., 91, 209. 22 Hammer, Ann. Physik, 33.445. 23 Becker, Ann. Physik, 1938, 32, 128. 2 4 Turnbull, J . Chenz. Physics, 1949, 17, 71.116 NUCLEATION AND GROWTH I N SUCROSE SOLUTIONS AF%=. the free energy of activation of viscosity, and x the niol fraction of solute. The net free energy required to form the nucleus is also A F = AFS - AF,, where AFs is the free energy required to form the surface, and AF, that gained in forming the mass of the crystal without any surface. Gibbs has shown that AFS = (3/2)AF?J ; whence A F = iAF,. If these reversible values are identified with the energies of activation of the respective processes, we have The energy of activation of viscosity is approximately 1/3 that of growth,ls so that as a crude approximation This relative order of magnitude has been pointed out before in an empirical waV.I8 AEnuc~eatiou = j@Ev.AEnucleation = (5/6)AEgowth. The Thomsen equation, 2cM ln(c/cco ) = c d ’ with A F = (1/3)AF, = (1/3) cr A = (43) n c y2 (as spheres), suggests at constant supersaturation. In these expressions c and c, are the solu- bilities at particle radii Y and CL) respectively, c the interfacial tension, A the surface, and M/d the molar volume. Since the activation energy is observed to increase with rising temperature at fixed interfacial conditions, it seems likely that some factor other than the work of forming the nucleus is involved in the nucleation process. Nothing definite is yet known about the entropy changes concerned in the above approximation, but the marked influence of stirring upon the rate of nucleation at higher concentrations is very suggestive of the viscosity as this factor.Surface Tension.-The interfacial tension, which is so prominent in most crystallization theories, has received special attention in the case of sucrose solution^.^ l3 l4 Since this interfacial tension between a solid and a liquid is difficult to evaluate, it has frequently been correlated with the ordinary surface tension of the liquid, although it seems questionable to specify it in this way. where s, I and g indicate solid, liquid and gas phases respectively, and 0 is the contact angle of wetting of the solid. If the wetting is complete, and the surface tension of the solid is constant, we have AF e 3 / ~ 2 Duprh’s rule for this type of interface is Osz = csg - OLg cos 8, Osl = cssg - czg, and dcsl = - dole The former is Antonoff’s rule for this case, and the latter indicates that ordinary surface-active materials, which usually decrease the liquid surface tension, may actually increase the interfacial tension at the solid surface. It was found impossible to increase the surface tension of crystallizable sucrose solutions to any extent by additives ; but ordinary wetting agents diminish it considerably.Even so, no great influence on the crystallizationANDREW VAN HOOK AND ARTHUR J. BRUNO 117 time was observed, which is contrary to several reports in the literature under similar circumstances.3 l 3 18 25 A twofold oversaturated solution of sucrose in 68 yo alcohol, whose surface tension was 26 dyneslcm., did not display crystals for 8 days, compared to about I day for an aqueous solution of the same supersaturation.Whether this prolongation is the result of lowered surface tension (and therefore possibly increased interfacial tension) or change in environment is not yet evident. These matters are being investigated further in this laboratory. Practical Implications .-The extreme difficulty of preparing and preserving supersaturated sucrose solutions would augur well for the applica- bility of the heterogeneous theory, in spite of the greater significance of the thermodynamic theory. Under conditions which prevail in the sugar house, as well as in ordinary laboratory work, nucleation undoubtedly occurs by chance inoculation. Under these circumstances, it is merely the rate of growth to visible size which determines the observed nucleation time. Since this process has been shown to be unimolecular,26 the observed equilateral hyperbola relation is an obvious one. However, as the concentration increases, a very strong and abrupt influence of true nucleation sets i n ; thus accounting for the metastable limits usually reported.27 Nucleation in condensed systems has all the attributes of a chain reaction,6 28 which feature explains the autocatalytic " false grain " region 27 of the sugar boiler. Conclusions .-The prominent features of the Volmer-Becker theory of nucleation are shown to be qualitatively applicable to supersaturated sucrose solutions. The continuing support of the Sugar Research Foundation, Inc., is gratefully acknowledged. Quantitative aspects will be investigated. Chemistry Department, CoUege of the Holy Cross, Worcester, Mass., U.S.A. z 5 Von Weimarn, 2. Chem. I n d . Koll., 1907, 2, 76 ; A.C.S. Absty., 3, 393. 26 Van Hook, I n d . Eng. Chem., 1944, 36, 1042. 27 Webre, PYOC. 11th Conf. Assoc. tec. azwcare'e7os Cuba, 197, p . 9. 28 Langmuir, PYOG. Amer. Phil. SOC., 1948, 92, 167.

ISSN:0366-9033    

DOI:10.1039/DF9490500112

出版商:RSC    

年代:1949

数据来源: RSC

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ANDREW VAN HOOK AND ARTHUR J. BRUNO 117 THE RELATIVE RATES OF GROWTH OF STRAINED AND UNSTRAINED AMMONIUM NITRATE CRYSTALS BY S. FORDHAM Received 4th February, 1949 At a discussion on crystallization held in Bristol in February, 1948, it was suggested by Prof. N. F. Mott that crystals should grow more rapidly when strained than when unstrained. During the course of a more general programme of work on the crystallization of ammonium nitrate, an opportunity was taken of testing this suggestion. Experimental The crystals of ammonium nitrate were prepared by slow evaporation a t It had been shown that Lissolamine A did not affect the crystallization 26' C .118 GROWTH OF AMMONIUM NITRATE CRYSTALS Mean Growth Original brcadth I mm. Strained 1 Unstrained ___ mm . 2'1 I2 2.080 0'451 1.235 1'347 0'444 1.408 1.500 0.352 1'594 1.685 0.130 1'467 1'5 72 0'125 1.596 1'693 0.189 1.396 I -42 8 0'1 19 1-208 1'221 0.091 1.292 1-318 0.126 of ammonium nitrate under such conditions, so the vessels were treated before the experiment with a dilute solution of this compound to prevent "creep." Some of the crystals which developed were prisms with good faces of the (110) form and usually with (011) faces, although the latter were frequently poorly developed.Specimens about 1-5 mm. broad and 4-5 mm. long were selected, roughly dried, and then shaken with a solution of Fixanol C in carbon tetra- chloride. The latter emulsified any adhering mother liquor and gave completely dry crystals, which were stored in a desiccator for use. Crystals were used within 24 hr.of preparation, during which time they retained quite adequate plasticity. The dry crystals were measured by means of a low-power microscope, and for each experiment two batches of five crystals each were selected, so as to be as nearly as possible equivalent in size and shape. Complete similarity was not, of course, attainable but in most cases variations in linear dimensions were within 10 yo. The crystals of one batch were strained by bending round a rod of 4 mm. diam., and their total weight found ; the other batch was weighed without straining. The strained and unstrained crystals were arranged Extra Growth I Ratio Of mm. , E x t g A L Y a n 0'010 I 0'022 0.019 0.043 0.105 ::::; 1 0.161 0.008 0.064 - 0'00 I - 0.005 0.032 0.269 0-0 I 7 0.187 0*026 , 0.206 TABLE I ~~ Mean growth is the average linear growth of all crystals.Extra growth is the amount by which the growth of strained crystals exceeded that of unstrained. alternately in a crystallizing dish, and allowed to grow in an ammonium nitrate solution evaporating at 26' C. After the required time the crystals were dried as before and the batches re-weighed. In the earlier trials growth by IOO yo in weight was reached, but in the later experiments this was reduced to 20-40 76, which was the minimum for which the experimental arrangements were suitable. From the measured weights, the mean breadths before and after growth were calculated on the assumption that the crystals were rectangular parallelepipeds, the lengths of the sides being in the ratio I : I : 3.The increase in breadth was used as a measure of growth, and was in most cases greater for the strained crystals. It will be seen that in all cases except one, the strained crystals grew more than the unstrained, but that the scatter of the recorded results was large. The standard deviation was calculated and the t-test applied to determine the significance of the mean, when it was found that the probability that the result was a chance variation from zero was about 0.002. It appears very likely, although not definitely proved, that strained crystals of ammonium nitrate grow faster than unstrained. Local variations in rate of growth undoubtedly occurred in these experiments, although their effect should have been eliminated by the method of analyzing the results.An attempt was made, however, to attain more uniform conditions by growing the crystals in a vertical tube with an air current sufficiently strong to maintain agitation. The trial was discontinued because fresh nucleation was extensive and results were very erratic. The results are given in Table I.S. FORDHAM 119 Discussion It appears most probable that ammonium nitrate grows faster from solution when strained. Such a statement, however, needs some elaboration before its true meaning becomes clear. The presence of strain should be shown at the surface by dislocations, and the increased rate of growth should be attributed to the presence of such dislocations and discontinuities. These experimental results do in fact support the theory that the normal growth of crystals is due to the propagation of dislocations.It would be expected that as the strained crystals grow, the number of effective extra dislocations remaining in the surface would diminish, so that the rate of growth of the two types of crystal should gradually become equal. There is, in fact, no significant correlation between the figures for " extra growth '' and " mean growth " recorded in Table I, and it is concluded that the effect of the straining had been eliminated before the shortest experi- ment was complete. Indeed, a formal rate may be calculated by taking the ratio of extra to mean growth, as in the last column of Table I, and it is found to have a negative regression coefficient on the mean growth, with significance 0.05-0.1. It would appear therefore that the method of straining used in these experiments caused dislocations which did not persist through a fresh layer 0.05 mm. thick, but which made crystallization more rapid in the early stages by at least 20 yo of its normal speed. Research Department, I.C.I. Limited, Nobel Division, Stevenston.

ISSN:0366-9033    

DOI:10.1039/DF9490500117

出版商:RSC    

年代:1949

数据来源: RSC

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S. FORDHAM CRYSTAL GROWTH FROM SOLUTION I. Layer Formation on Crystal Faces BY c. w. BUNN AND H. EMMETT Received 20th January, 1949 In this and the following paper, the results of many observations and experiments made over a number of years (since 1932) are collected and discussed. The work on crystal growth from solution which has been carried out in this laboratory from time to time, as opportunity occurred, was started tvith the object of discovering how the rates of growth of crystal faces are related to the structure of the crystal face and the concentration of solute in the surrounding solution. It was at first hoped that the problem could be treated in terms of simple physico-chemical concepts : the rate of growth of a particular face was perhaps some function of the supersaturation of the solution in contact with the face, while different faces of the same crystal or of crystals of different substances would be characterized by different constants which would depend on the surface forces. Such relations have usually been assumed in previous theoretical speculations, such as those of Berthoud and Valeton.2 But it became evident that,.on the one hand, there was no correlation between the rate of growth of a particular type of face and the supersaturation at the face; 1 Berthoud, J.Chim. Physique, 1912, 10, 624. 2 Valeton, 2. Krist., 1924, 59, 135, 335.I20 CRYSTAL GROWTH FROM SOLUTION and on the other, that a growing crystal face is not a uniform surface, and that surface forces must vary at different points. The work then became frankly exploratory, and as far as possible uninfluenced by preconceived hypotheses.The surfaces of growing crystals of many substances were observed as closely as possible under the microscope, in order to learn as much as possible about the manner of deposition of solute and the fine structure of growing faces ; this work is reported in the present paper. The other aspect of the problem-the concentration of solution and its variation round a growing crystal-is dealt with in the second paper. The polyhedral habit of most crystals suggests that material is deposited on the faces in successive layers; and the theoretical work of Ko~sel,~ Stranski * and Brandes and Volmer supports the idea that, at any rate for ionic crystals of NaCl type of structure, this is the manner in which ions are built on to the crystal : the energy yield when an ion is added to an incomplete layer is greater than that for the starting of a new layer, and therefore there is a tendency for a layer, once started, to be completed rapidly, the inception of the next layer being delayed.Direct experimental evidence on the matter, on the ionic or molecular scale, is not available ; but layers a few molecules thick have been detected on crystals of m-toluidine by Marcelin and by Kowarski.7 This substance grows from alcoholic solution in extremely thin plates, thin enough to give interference colours like those seen in oil films on water ; discontinuities in the shade of inter- ference colour, which meant discontinuities of thickness, could be seen moving across the crystal faces, and it was calculated that in some circum- stances the layers thus revealed were only a few tens of 5ngstrom-in fact, only a few molecules-in thickness.Similar thin layers were observed by Volmer on thin crystals of PbI, formed by mixing solutions of Pb(NO,), and KI. Not many substances grow in sufficiently thin plates to give this type of evidence; but we found that thick crystals of some substances, when observed at high power, using dark ground illumination, show layers spreading across the faces, and these, to be visible at all, must be very much thicker than those seen on m-toluidine or PbI,; they must be at least several hundred ingstrom thick. On a few crystals the layers are so thick that they can be seen either in ordinary transmitted light or in bire- fringent crystals by using crossed Nicols.I t was possible to make many observations of these layers on a variety of different crystals, to observe their point of origin, to measure their thickness and rate of spreading, and the effect on them of the presence of dissolved impurities and of different solvents. Most of the work has been qualitative, and constitutes an extensive superficial survey of the phenomenon of layer formation on crystals of many different substances. In all the experiments, a drop of warm saturated solution was placed on a warm microscope slide, covered with a thin cover- slip, and observed while cooling on the microscope stage. A cardioid condenser was used to give dark-ground illumination at high powers.The outstanding generalizations which emerged from these observations are the following- I. Layers very often start, not from edges or corners of crystals, but from the centres of faces, spreading outwards towards the edges. a Kossel, Nach. Ges. Wiss. Gbttingen, 1927, 135 ; Metallwirtschaft., 1929, 8, 877 ; Natuvwiss., 1930, 18, 901. Stranski, 2. physik. Chem., 1928, 136, 259. Brandes and Volmer, 2. physik. Chem. A, 1931, 155, 466. Marcelin, Ann. Physique, 1918, 10, 185. Kowarski, J . China. Physique, 1935, 32, 303, 395, 469. Volmer, 2. $hysiR. Chem., 1923, 102, 267.a a b b C C d - d e f 171G. I. FIG. 2. F I G . 3. FIG. I.--SaCl. Photographs at I-sec. intervals. ( x 300) F i G . 2.-NaCl. g-sec. interval.. Pit formed by layer encirclement, and subsequently filled in.( x 245) FIG. 3.-KH,PO, I-sec. intervals. ( x 35) To face page 12 11C. W. BUNN AND H. EMMETT I21 2. The thickness of the layers on many crystals increases as the layers approach the edges of the crystal faces. 3. The boundaries of the layers are often irregular, especially when growth is rapid; but as growth slows down there is a tendency to regularity of shape, the actual shape conforming to the symmetry of the crystal face. 4. Dissolved impurities may strongly influence the thickness and the shape of the layers, the effect being highly specific. 5. Thick layers have been seen only on crystals of certain ionic or polar substances ; they have not been seen, under the experimental conditions used, on crystals of non-polar substances. Experiment a1 Sodium Chloride .-The first four of the above generalizations are well illustrated by the phenomena observed in experiments with sodium chloride.The cubic crystals usually lie on a microscope slide on a cube face, so that one face is seen normally and four others edgewise. On the face seen normally, layers were usually observed to be spreading outwards from a point roughly in the centre of the face. It was not possible to locate the point exactly; nothing could be seen a t the centre of the system of layers, and the layers only became visible at a distance from the centre which varied considerably but was often one-quarter to one-half of the distance to the edge. From the increasing plainness of the layer boundaries towards the edges of a face, it appeared that the layers increased in thickness as they spread outwards.The layers were usually rather irregular in shape, though sometimes suggesting a square ; but the addition of I % CaCl, to the solution had the effect of making the layers more regular in shape, roughly octagonal, and more easily visible. I t was possible to obtain cinematograph rezords of the growth of the layers under these circumstances. (A Pathe camera (9.5 mm. film), with the lens removed, was used at normal speed. The microscope objective (2.9 mm. oil immersion) cast an image straight on to the film. For further details see paper by Emmett.Q) Four shots selected from the cinematograph film are shown in Fig. I a-d ; they show the same face at successive intervals of I second. A new layer, which is barely visible in I b, is easily visible in I c and has spread considerably in I d.On the faces seen edgewise, i t was sometimes possible to see the thickness of the layers, but in general, in the case of sodium chloride, the layers were so thin that they showed up best on the face seen normally, owing to the scattering light by the edge of each layer. (By ordinary transmitted light the layers could not be seen at all.) The average thickness of the layers was measured by first focusing on a particular point on the surface, counting the passage of some 30-50 layers past the point, refocusing and reading off the change in position of the calibrated fine adjustment screw of the microscope. After correcting for the refractive index of the solution, the total thickness of the layers was obtained, and hence the average thickness of one layer.In several ex eriments, the layer thickness cell edge of sodium chloride is 5-63 A, each layer edge is a wall 300-700 unit cells high (or twice this number of atoms). The rate of spreading of the layers in one experiment was found by measuring a cinematograph film. All the layers spread at about the same uniform speed, the rate of advance of an edge (in the direction of a cube edge) being 2-5 X 10-* cm./sec. ; the rate declined a little as time went on. (The rate of spreading, like the thickness, varied considerably with the specimen of salt and the conditions of the experiment, and this figure by itself has no great significance ; it is quoted simply to give an idea of the order of magnitude.) The rate of spreading of a layer diminishes slightly as i t spreads outwards.The effect of CaC1, in making the layers thicker and more regular has already been was found to be 2800, 3500, 4100, 3600 and 1700 i . As the length of the unit Certain impurities affected the growth of the layers quite profoundly. @ Emmett, J . Micro. SOC., 1943, 63, 26.I22 CRYSTAL GROWTH FROM SOLUTION mentioned ; Na,SbO, had a similar effect ; but most other substances which were tied either had the reverse effect-that is, they made the layers thinner- or else had no marked effect. The following substances had the effect of making the layers thinner ; and, if added in sufficient concentration, gave NaCl crystals which showed no sign of layers. The percentage figure given is the concentration required to give clear crystals showing no layers : Pb(N03), (0’05 %), PbC1, (0.05 %), Bi(NOJ3 (0-1 yo), BiCl, (0.1 %), MnCl, (2.0 %), CdCl, (0.02 %), SrC1, (2.0 yo), SnCl, (NZ %).The following substances appeared to have no appreciable effect- HCI, Na,SO,, NiCl,, CoCl,, SbCl,, A1 (NO,),, BaCl,, MgCl,, ZnCl,, FeCl,, LiC1, PtCl,, HgCl,, and cerous nitrate. The effect of impurities on the formation of layers is thus specific, and recalls the modification of crystal shape by dissolved impurities, which is also a highly specific effect. But note that the only sub- stances in the above list which caused the appearance of other than cube faces were B,(NO,), and BiCI3, which produced (110) as well as (100) faces. In the great majority of experiments, only one system of layers was seen on any one face ; but on a few occasions two systems, spreading from different points, were seen ; and on one occasion, when an unusually large crystal was observed, no less than five systems were seen. In all cases the points from which the layers spread (whether there was one centre or more on a face) were not on the corners or edges, but well within the boundaries of the face ; and when there was only one point, i t was roughly a t the centre of the face.Another phenomenon seen occasionally was the formation of a pit by encircle- ment of a small area by a layer growing all round it. This is well shown by the series of photographs in Fig. 2, in which a pit forms towards each end of a long rectangular face on which a layer system is spreading from the centre ; these pits are soon filled in.On this occasion the process of formation and subsequent filling-in of a pit was repeated several times in rapid succession. The phenomenon recalls the triangular pits on diamond crystals reported by Tolansky and Wilcock 10 and believed to have arisen during growth ; the actual growth of diamond cannot be observed, but the present example (and many others which have been seen on various crystals) shows that layer encirclement does occur and lends support to Tolansky and Wilcock’s explanation. Cadmium Iodide.-This substance grows in the form of hexagonal plates ; if the basal planes are observed when the crystals are growing from aqueous solution, layers can be seen spreading over the faces, even in ordinary transmitted light.The layers observed in these experiments were roughly circular in shape, or occasionally vaguely hexagonal. Fig. 4 shows a typical crystal with a single system of layers spreading outwards from a point which can be located more precisely than in the case of sodium chloride ; the layers can be seen much nearer to the origin. Fig. 5 shows an example in which there are two origins, and the layers are more nearly hexagonal in shape. The layers on this substance were usually a little thicker than those on sodium chloride-from 3000 to 5000 A ; the reason why they are often more easily visible than those on sodium chloride is partly that they are thicker, but chiefly because of the high refractive index of CdI,. The rate of spreading was about one-fifth that of sodium chloride.Potassium Dihydrogen Phosphate .-This is another substance on which layers can often be seen without the aid of dark ground illumination. It is tetragonal, and grows in the form of prisms of square cross-section terminated by pyramids, and the crystals usually lie on the microscope slide so that the long rectangular (110) prism faces are seen normally. On these faces, layers can be seen spreading outwards from a point which is, more often than not, roughly in the centre of the face. The layers are sometimes rectangular, as in Fig. 6, but are more often very irregular in shape. A phenomenon which this substance shows particularly clearly is that of very thin rapidly advancing layers overtaking thicker and more slowly advancing ones ; this appears to be the way in which thick layers are built up, and if we may imagine the same 10 Tolansky and Wilcock, Nature, 1946, 157, 583.F I G .4. FIG. 5. FIG. 6. FIG. 7. FIG. 4 and 5.-Cd12. ( X 190) FIG. 6 and 7.-KH2PO,. Polarized light, crossed Nicols. ( x 150) [To face page 122C . W. BUNN AND H. EMMETT process occurring on a smaller scale, it is probably in this way that layers of visible thickness are built up from original layers which are perhaps one ion or a few ions thick. The increasing thickness of layers as they grow outwards is also shown strikingly by this substance ; note that in Fig. 3 no layers can be seen a t the centre, but they become very plain towards the ends of the crystal ; when the cinematograph film is watched, each layer gradually becomes visible as i t grows outwards, and this is probably due to the overtaking of much thinner layers, too thin to be seen.Another phenomenon which occurs in Fig. 3 is the formation of a pit by encirclement of a small area by a layer growing all round it. The beginning of encirclement is seen on the right-hand side of 3 d (see arrow) ; the pit is clearly visible in 3 e (arrow) ; a t 3 f i t has disappeared, having been filled in by inward growth of the layer. One more observation on Fig. 3 : the confused appearance of the bottom left-hand corner of each photograph is due to the existence of a number of irregular layers, some of which overtake others ; the sequence of events can only be properly appreciated by watching the actual process or the cinematograph record.The addition of phosphoric acid to the solution had the effect of making the layers thinner and more difficult to see; when K,HP04 was added to the solution no layers could be seen a t all. \ FIG. 8.-Layers “ wrapping over ” from one face to another. (Octahedral faces of Pb(NO,),.) On crystals of this substance, the layers are often well seen when crossed Njcols are used ; the interference colour arising from the birefringence of the crystal varies in shade according to the thickness, and the colour effects show up the growth of the layers in a beautiful manner, though appearances are sometimes confused, owing to the fact that two systems of layers, one above and one below the crystal, are seen simultaneously. Some very striking cinematograph shots (in colour) were secured under these conditions.Fig. 7 is a monochrome “still” from one of these films; it shows, in the upper part, numerous thin irregular-shaped layers, and in the lower part, some more regular formations. This photograph also shows other striking characteristics of this substance, when grown rapidly on a microscope slide-the tendency for different parts of the same rod-like crystal to grow almost independently (note the contrast between the two layer systems in the two halves), and the tendency to form very thick sheath-like layers, embracing all four sides of the tetragonal prism, which grow slowly along the prism. Other Salts.-Many other salts were observed, and in the majority of cases no layers were seen ; but several other crystals were found to exhibit visible layers.Among them was lead nitrate, the normal habit of which is octahedral. The layers which grew on the triangular faces were roughly triangular in shape, when they grew from a point in the centre of a face. Sometimes, on the other hand, layers seemed to be spreading from a corner or an edge ; the origin of those spreading from an edge was, perhaps, indicated by occasional observations that a layer, on reaching an edge, wrapped over on to the next face, as in Fig. 8. The addition of Methylene Blue to the solution caused the crystals to grow as cubes; on these also layers were seen, usually apparently growingI24 CRYSTAL GROWTH FROM SOLUTION from corners and edges ; but whether the corners and edges were the real origin of the layers is doubtful, in view of the " wrapping-over " effect just mentioned.When a layer system can be seen spreading from the centre of a face, there is no doubt about the origin, but when i t appears to come from an edge or corner, it may be a legacy from another face, which cannot be examined. The addition of sodium nitrate to the solution had no effect, but when nitric acid was added no layers could be seen. When both sodium nitrate and nitric acid were added layers were again seen. Sodium nitrate, growing from pure solution, showed no layers, but the addition of lead nitrate to the solution caused layers to appear. No other nitrates had any effect when added to NaNO, solution ; those of Ca, Sr, Ba, Cu, Ni, Ag, Hg, Bi and A1 as well as nitric acid were tried.Potassium sulphate and potassium chromate showed extremely thin layers. The addition of H,SO,, NaOH, KOH, or Na,SO, to a solution of potassium sulphate did not affect the formation of layers. Alum showed slight indications of layers. FeSO,. (NH3, SO,. 6H,O showed clearly defined layers, and it was on one of these crystals that a system of layers was seen growing inwards (filling a pit), on the same face with a normal outward-growing system (see Fig. 9) ; the inward-growing system could be seen to be due partly to the wrapping-over of layers from contiguous faces. FIG. 9.-Outward-growing and inward-growing layer systems on the same face of a Note the part played by the " wrapping- crystal of ferrous ammonium sulphate. over " effect. Potassium iodide showed some confused appearances which might be interpreted as layer formation ; these were seen near edges and corners, but the direction of spreading was not clear.Mercuric chloride showed definite layers ; so did magnesium nitrate. Sodium formate, sodium diethyldi thiocarbamate and sodium phthalate showed very thin layers. Layers could not be observed on any of the following crystals-NaClO,, NaIO,, NaF, NaBrO,, NaNO,, NaNO,, sodium antimonate and sodium nitro- prusside ; KNO,, KNO,, KC103, K,Cr,O,, KSCN, KClO,, K,S,O, ; sodium citrate, potassium oxalate and tetraoxalate ; ammonium chloride, either from pure solution or a solution containing urea ; LiC1, ZnSO, , 7H,O , MgSO, . 7H,O, (NH,),SO,, ammonium vanadate, chrome alum ; and NaHNH,PO, .4H,O. All those substances on which layers were seen have moderate or high solu- bilities (the lowest being HgC1, with a solubility of about 6 yo at zoo C).The following slightly soluble substances were observed, but no layers were seen on any of them- CaSO,. zH,O, CaC1,. Ca(OH), . H,O, 2Na,PO,. NaF . 1gH,O, Na,B,O, , 1oH,O, PbCl,, AgSO,, Li2C03, Sr(OH),, Ba(OH),, KClO,. Organic Substances .-The following crystals have been observed, but n o layers have ever been seen under the conditions of these experiments- Cane sugar, hydroquinone, propionamide, phthalic acid, citric acid (in water), naphthalene, benzophenone, camphor, bend, stilbene, p-nitrophenol, methyl- glyoxime, dimidone, dichlorobenzene, acetanilide and a-naphthol (in alcohol) ; and anthracene in benzene solution.C.W. BUNN AND H. EMMETT 125 On the other hand, urea, acetamide, pyrocatechol and chloramine-T (CH, . C,H,SO,Na . HC1.3H2O) showed definite layers, when grown from aqueous solution ; so also did sodium formate, sodium diethyldithiocarbamate and sodium phthalate. Note that all the substances on which layers were seen are either ionic or else contain strongly polar groups. Influence of Solvent.-It appeared from all the foregoing observations that layers thick enough to be seen by ordinary light under the conditions of these experiments are only formed on crystals which contain either ions or strongly polar groups ; in view of this apparent influence of electrostatic forces, it seemed of interest to study the growth of crystals in solvents of lower dielectric constant than water.Several substances which had been observed to give visible layers when grown from aqueous solution, and which are also soluble in ethyl alcohol, were therefore crystallized from this solvent. These substances were cadmium iodide, potassium dihydrogen phosphate, sodium formate, sodium diethyldithiocarbamate, urea, acetamide, pyrocatechol and chloramine-T. In all cases, layers did not appear on crystals growing from alcoholic solution. In three cases-urea, potassium dihydrogen phosphate and cadmium iodide- alcohol-water mixtures were also used, and it was evident that the layers became thinner with increasing proportion of alcohol. When urea was grown from methyl alcohol layers were seen, as in water solution ; but cadmium iodide in methyl alcohol showed no layers.I t seems reasonable to attribute these effects to the fact that the dielectric constant of ethyl alcohol (26 at 2oOC) is much lower than that of water (81 at 18" C) ; the dielectric constant of methyl alcohol (31 a t zoo C) is a little higher than that of ethyl alcohol. Discussion Formation of Thick Layers.-It seems likely that crystals in general (at any rate, those with definite faces) grow by the spreading of discrete layers one after another across the faces. On many crystals these layers are too thin to be seen by visible light ; but, as we have seen, on quite a number of crystals the layers are sufficiently thick to be seen either by dark ground illumination or sometimes even by ordinary transmitted light or between crossed Nicols; these layers are often some thousands of 5ngstrom in thickness.That the layer-spreading process occurs on the much smaller scale of a few molecules has been shown by the observations of Marcelin 6 and Kowarski 7 on fi-toluidine, which, because it grows as exceedingly thin plates, permits observations by a method sensitive to much smaller differences of thickness than those revealed under the present conditions; and the process of formation of thick layers from thin ones is demonstrated by our frequent observations that thin layers, spreading more rapidly than thicker ones, overtake underlying thicker layers and add to their thickness. Why are thick layers, hundreds of ions or molecules in thickness, built up ? Why do not the thin primary layers, which are perhaps one ion or molecule, or a few ions or molecules, in thickness, proceed independently ? It might be urged that thin layers spread faster than thick ones, simply because less solute is required to extend a thin layer a certain distance than to extend a thick layer to the same distance. But this is beside the point : a system of thin layers of a certain height needs precisely the same amount of solute to spread a certain distance as a system of fewer thick layers of the same total height.What we have to explain is the tendency of a system of a large number of thin layers to break up into a system of a few thick layers. Although no quantitative theory based on a consideration of surface forces can be offered, we can at any rate link up the phenomenon of thick layer formation with the general principles of crystal morphology. When a succession of thin layers, of ionic or molecular thickness, is spreading across a crystal face, the surface (see Fig.~ o a ) is not a face of low indices but a face having very high indices. Now the outstanding generalization126 CRYSTAL GROWTH FROM SOLUTION of crystal morphology is the universal tendency for the bounding surfaces to be faces of low indices; when growth comes to an end the faces are found to be those of low indices, often indeed the simplest possible indices; these are the faces which are either parallel to, or are simply related to, the edges of the unit cell. Faces with high indices, if they are artificially created by cutting or dissolving a crystal, eliminate themselves because their rates of growth (thickness deposited on the surface in unit time) are higher than those of simple faces.The elimination may occur by apparently straightforward growth as in Fig. 11 a or by a process of breaking-up into steps as in Fig. 11 b. Faces of high indices are thus less 'r ............. ....................... ........................... ........................... ........................... ........................... ....... ....... ....... ....... A dva M c i ~ y ~ / /aye / A I ........ .Ic- ............... ............... mole c ula ?-5 \ I 1 Y .... ........... r ... Low- index, 4 SUYfGCc2 '\ Advancihy rhi'ck --,..;.I ............... L' _-- ................ .....k.'-- L y f . ..................... ..................... ..................... .......................................... ................................ c . .................................... .................................... .................................... .................................... ..................... -* (= FIG. 10 (a) A system of molecular layers spreading, one after another, across a face ( b ) The break-up into large steps has the effect of making the major part of (c) Idealized representation of the surface of a growing crystal. (Heights of constitutes a high-index surface. the surface a low-index surface. steps much exaggerated.) stable as surfaces, and presumably have a higher surface energy, than those of low indices. There will therefore be a tendency for the high-index faces created by the system of spreading layers (Fig.roa) to break up into comparatively large steps, so that the bounding surfaces have low indices (Fig. ~ o b ) . (Actually the edge of a step is usually a high-index surface (see below) ; nevertheless, the break-up into large steps does have the effect of making the great majority of the surface (the tops of the layers) a low-index surface.) There is, of course, no sharp distinction between the original surface which can appropriately be regarded as a high-index face (Fig. roa) and the surface with large steps (Fig. ~ o b ) which is pre- dominantly a low-index surface; there is a continuous change of surface energy with step-height. Thus the reason why thin layers spread moreC. W. BUNN AND H. EMMETT rapidly than thicker ones, and therefore overtake thicker ones, is in all probability that there is a greater surface energy at a low step than at a high one.The formation of thick layers hundreds of ions or molecules thick is thus seen to be just another manifestation of the great morphological principle of simple indices. The “ vicinal ” faces which have been found on minerals and on crystals grown in the laboratory probably consist of systems of layers having a step-height smaller than the wavelength of light ; the crystals exhibiting vicinal faces are probably crystals whose growth has not gradually slowed down with time, but has for some reason been arrested. The existence of vicinal faces on growing crystals was shown by Miers (I903), who also observed that the inclination varied during growth.The crystals he studied were alum, sodium chlorate, ZnSO, .7H,O and MgSO, .7H,O ; all these have been studied in the present work, and only on alum have any indications of layers been seen ; but if our interpretation of vicinal faces is correct, Miers’ observations may be taken as evidence that layers do form on the other two crystals but are too thin to be seen. FIG. I I.-Elimination of high-index surface, (a) by apparently straightforward growth (the rate of growth of the high-index face being greater than those of neighbouring low-index faces), ( b ) by step formation. The layer thickness built up on any crystal presumably depends on the difference between the surface energies of high-index and low-index faces (in relation to the surrounding solution), which may perhaps be formulated for the present circumstances as the change of surface energy with step- height ; our observations have shown that this is different for each crystal, and, moreover, is strongly influenced by specific impurities in the solution. The only generalization we can make is that since thick layers were never seen on crystals of non-polar substances, but were seen on a number of crystals containing ions or polar molecules, the change of surface energy with step-height is steeper in ionic or polar crystals than in non-polar ones.It is entirely reasonable that this should be so ; in a crystal composed of ions or polar molecules, different faces present very different arrangements of positive and negative charges, and the surface forces would be expected to vary sharply with the distribution of surface charges ; but in non-polar molecular crystals there are no sharp electrostatic differences, and it is to be expected that the differences between the surface forces of different faces would be less marked. The differences between various ionic and polar crystals, some of which show thick layers, while others do not, remain unexplained; we have not been able to detect any correlation between chemical constitution and the presence or absence of thick layers.In view of the powerful influence of dissolved impurities, any relationship could only be expected to be found if highly purified substances were used. The manner of deposition on the edge of a layer must also be considered. A layer edge several hundred atoms high is quite a large face, from the atomic point of view ; does deposition occur on it by the formation of surface128 CRYSTAL GROWTH FROM SOLUTION nuclei and the subsequent spreading of layers on a smaller scale than that involved in the thick layers, or does solute pile on in a more indiscriminate manner ? The answer to this question is probably bound up with the question of the nature of the surface of a layer edge.Most growing layers are irregular in shape and even if (on a cubic crystal) the edges are at right-angles to the tops, the edges are, for the most part, not low-index surfaces. (This is symbolized in Fig. IOC by making the edges of layers non-rectangular.) Deposition on the edges of layers, therefore, is deposition on high-index surfaces.There is some reason for thinking that rapid deposition on high-index surfaces does not occur by layer formation, but in a more indiscriminate way. The skeletal shapes of very rapidly growing crystals usually have rounded surfaces ; the directions of growth are well defined geometrically, but the actual surfaces are rounded. (Flat surfaces may develop subse- quently, and these are low-index surfaces, but during rapid growth the surfaces are rounded.) The absence of flat surfaces during growth suggests that deposition occurs, not by layer formation, but in a more indiscriminate manner. Therefore deposition on the edges of layers may also take place in this way. There is, of course, a tendency for the edges of layers to become low-index surfaces; this is seen, first of all, as a tendency for the shape of a layer to become more regular as growth slows down; but even when the shape becomes fairly regular, as in the octagonal layers of Fig.I (NaCl), it may be doubted whether the edges are surfaces of minimum indices for this reason. If the edges of the above-mentioned octagonal layers were per- pendicular to the top (a (100) face, let us say), the surfaces of the edges would be (010) and (011) ; a (010) face is crystallographically equivalent to a (100) face, and deposition would therefore be equally likely on the edge and the top of the layer ; that deposition occurred only or mainly on the edge makes one doubt whether the edge was really (0x0) ; it was probably not at right-angles to the top and was therefore a high-index face.Nevertheless, as growth slows down, there is presumably a tendency for the edges of layers to become low-index surfaces. It may be observed that when this happens growth will be very much inhibited, for it will have to wait on the formation of surface nuclei. It is possible that it is in these circumstances the Kossel-Stranski picture of surface nuclei forming on edges and corners is valid; we may well imagine that the most likely place for a nucleus to form is at the edge formed by the top and side of a layer (A in Fig. ~ o b ) . We are thus led to the view that rapid growth of crystals depends on the maintenance of high-index surfaces; if for any reason the surface (so to speak) heals-that is, becomes a low-index surface-growth will be very much inhibited.We shall return to the question of the rate of growth of crystals in Part 11. Formation of Surface Nuclei.-Perhaps the most striking and the most important generalization which came out of the many observations made in this work is that, more often than not, the layers were observed to spread, not from the corners or edges of faces, but from the centres of faces. Even when layers were seen spreading from edges or corners, it was sometimes evident that they had wrapped-over ” from contiguous faces. One example may be quoted in which it appeared fairly certain that the point of origin was a corner: on a lead nitrate crystal, a small, rapidly growing face, which showed normal layer growth by spreading from the centre, became smaller and smaller owing to its rapid growth compared with contiguous faces, and was soon eliminated and replaced by a corner; immediately this happened, layers were seen spreading from And if layers are formed, where do they start ?C.W. BUNN AND H. EMMETT 129 this corner over the only contiguous face which could be observed clearly (see Fig. 12). The sequence of events suggests strongly that in these circumstances the layers really did originate at the corner and were not " wrapped-over " from other faces. Nevertheless, the general rule is that layers spread from the centres of faces. FIG. Iz.-Layers spreading from corner of octahedral face of Pb(N0,) .a crystal, immedi- ately following the disappearance of a small, rapidly growing face. E130 CRYSTAL GROWTH FROM SOLUTION To the observations made in this work we may add one more, which, although not an observation of growing crystal but a record of the surface structure of a crystal after growth was arrested, seems convincing.Electron microscope photographs taken by R. W. G. Wyckofi (private communication) show, on the surfaces of crystals of a protein (a decomposition product of one of the necrosis viruses), low pyramids of layers. Individual molecules 130 in diameter can be seen, and it appears that each layer is one molecule thick ; further, the concentric arrangement suggests that during growth layers spread out successively from a point which is not on a crystal edge or corner, just as in the examples seen in the present work, This confirms that, as we have already surmised, the layer-spreading process which we have observed on the scale of hundreds or thousands of Bngstrom gives a correct impression of what happens on the molecular scale.The spreading of layers from the centres of crystal faces rather than from edges or corners seems surprising, whether one approaches the problem from a consideration of surface forces on the crystal or of the conditions in the solution. If we consider the crystal, the edges and corners are the places where there are most unsatisfied forces, and these are the places where nuclei for the inception of new layers would be expected to form. This idea seems quite generally applicable to all crystals ; moreover, in the case of ionic crystals having the NaCl structure, Kossel,3 Stranski and Brandes and Volmer have calculated the energy-yield in adding an ion to various places on a crystal surface (an atomically perfect plane, the edge of an incomplete layer, etc.) and their calculations indicate that, if we assume the event of greatest energy-yield to be statistically preferred, then the inception of new layers is more likely to occur at edges or corners than in the centre of a face.If we consider the solution surrounding the crystal, it has been shown (see Part 11, and Berg 11) that the supersaturation is greater at the edges of a face than at the centre, and it is at the places where supersaturation is highest that we should expect new surface nuclei to form. Thus, whether we consider the crystal surface or the solution in contact with the crystal, we are led by current ideas to expect that new layers would start at the edges and corners.But, as we have seen, layers usually do not spread from edges or corners, but from the centres of faces. It is evident that current ideas need revision, or else there is some other factor which overshadows those which have so far been considered. What are the circumstances at the centre of a face, other than those already considered? If we consider the structure of the crystal surface, a possible cause of the formation of surface nuclei is the existence of cracks or strains; it is noteworthy that when sodium nitrate crystals grow on a calcite cleavage, they grow more freely on cracks than elsewhere; but we know of no evidence indicating (as a general rule) greater imperfection at the centre of a crystal face than elsewhere. If we turn to the solution, a possible clue is given by the study of the concentration distribution round growing crystals of sodium chlorate ; according to Berg l1 less solute arrives, per unit area of face, at the edges of a face than in the centre, and therefore, since faces remain nearly flat, surface migration of solute molecules must take place from the centre towards the edges of a face.(In our own earlier work on this aspect, which is considered in Part 11, we could not be sure that this was so; but Berg’s measurements were perhaps rather more precise.) If it is a fact that the amount of solute per unit area arriving at the centre of a face is greater than at the edges, this may be the reason 11Be1-g Proc. Roy. SOC. A , 1938, 164, 79.Cp. also Humphreys-Owen, Proc. Roy. SOC. A (in press) and This Discussion.C . W. BUNN AND H. EMMETT why layers start at the centre ; in spite of the surface migration towards the corners, which tends to relieve the situation, the piling-up of excess solute at the centre is likely to result in additional deposition there-that is, in the formation of surface nuclei. It is suggested in Part I1 that the tendency for excess solute to arrive at the centre of a face is due to the geometry of the situation : radial inward diffusion to a polygonal crystal necessarily tends to deliver excess solute to the centres of crystal faces. I t is not possible to decide with certainty how far the formation of surface nuclei at the centre of a face is due to surface structure and how far to the disposition of concentration gradients in the solution.The occasional observation of more than one system of layers on a face would appear to favour surface imperfections, but it is not impossible even in these cases that diffusion effects were responsible : convection currents might lead to more than one point of convergence of excess solute on the same face. Layer Formation and the Imperfections of Crystals.-There is a great deal of evidence which indicates that most crystals, even those which are perfectly transparent and have highly perfect faces, are very imperfect in structure. The tensile strengths of actual crystals are only small fractions of what they would be for perfect crystals; and the intensities of X-ray reflections indicate that the precise structure which exists in small regions is not continued uninterrupted throughout the crystal-there are dis- continuities at intervals of the order of IOOOA.The discontinuities are not at regular intervals; the idea of a regular secondary structure due to fundamental causes, which was at one time put forward by Zwicky12 is not now accepted ; there are considerable variations in imperfection of crystals from different sources, as Smekal l3 has shown by measuring mech- anical properties of rock-salt crystals, and Lonsdale l4 by divergent-beam X-ray photography of various crystals. It seems likely that these imper- fections arise (at any rate partly) from the manner of growth by the spreading of layers across the faces. Successive layers do not necessarily join up perfectly with each other; it is more likely that cracks will occur, and, moreover, we have occasionally seen on urea crystals layers which, starting apparently in contact with the underlying solid, actually part company with it, leaving a visible crack.Another fact pointing in the same direction is that crystals on which no layers have been seen (that is, on which the layers are too thin to be seen) tend to be more perfectly transparent than those on which thick layers have been seen, These are extreme cases, but the same sort of thing is likely to proceed on a smaller, invisible scale. When layers fail to join up properly, there will be not only a crack between them but also small changes of orientation of the lattice, perhaps best visualized as a slight waviness of each layer. Both types of imperfection are necessary to account for the intensities of X-ray reflections. Layers, to be visible a t all under the conditions of our observations, must be a t least several hundred Angstrom thick; and those layers whose thickness was measured were 1700-5000 A thick. For many substances, the layers are on the border-line of visibility, while for many others, they are too thin to be seen. The order of magnitude is about right to account for the fact that many crystals are, as far as X-ray diffraction is concerned, “ ideally imperfect.” We have also seen that the thickness of layers is strongly influenced by specific dissolved impurities and the nature of the solvent. There would appear to be scope for investigating the intensities l2 Zwicky, Proc. Nut. Acad. Sci., 1929, 15, 253 ; Physic. Rev., 1931, 38, 1772 ; l3 Smekal, Physik. Z., 1930, 31, 229. l4 Lonsdale, Phil. Trans. Roy. Soc., 1947, 240, 319. 1932, 401 63.I32 CRYSTAL GROWTH FROM SOLUTION of X-ray reflections of crystals whose growth has previously been observed ; there might be a correlation between layer thickness during growth and the " extinction " effects often found in X-ray reflections ; the layer thickness on urea crystals, for instance, could be varied by growing from mixtures of different proportions of alcohol and water. Again, on many crystals, including NaCl and KH,PO,, the layers are thicker towards the edges of a face than at its centre ; there might be a difference between the intensities of reflection of X-rays at these two positions on the crystal face. I.C.I. Lta., Alkali Division, Northwich, Cheshire.

ISSN:0366-9033    

DOI:10.1039/DF9490500119

出版商:RSC    

年代:1949

数据来源: RSC

摘要:

CRYSTAL GROWTH FROM SOLUTION CRYSTAL GROWTH FROM SOLUTION 11. Concentration Gradients and the Rates of Growth of Crystals BY C. W. BUNN Received 31st January, 1949 When a crystal grows from a supersaturated solution, the concentration of the solution in contact with the crystal is reduced, a concentration gradient is set up, and the crystal is fed by diffusion down this gradient. We may divide the process of crystal growth into two aspects-the " taking " of solute from the solution by the crystal face, and the arrival of more solute by diffusion ; and the rate of growth of a crystal face (thickness deposited in unit time) depends on the factors which control both aspects. A knowledge of the supersaturation at the face is necessary for the consideration of both aspects, for on the one hand it may control the rate at which solute can be " taken " by the face, and on the other, the concentration gradients which are set up depend on the supersaturation at the face as well as on the initial supersaturation of the solution.The diffusive flow of solute is governed by Fick's law, that the rate of diffusion past any point is propor- tional to the concentration gradient at that point. (Deviations from Fick's law-that is, variations of diffusion constant with concentration l---need not concern us at present.) that the concentration at the crystal face sinks to the solubility value; but in 1903 M i e r ~ , ~ by measuring the angle of total internal reflection at a growing crystal face, determined the refractive index of the solution and hence its composition, and established that the solution at the face is very appreciably supersaturated; he found this was so for three substances-alum, sodium chlorate and sodium nitrate.In later theoretical speculations, such as those of Berthoud,* Valeton 5 and SpangenbergB the prevalent idea has been that the rate of growth of a crystal face of a given type is some function of the supersaturation at the face, the underlying conception being that the supersaturation is a measure It was at one time supposed 1 McBain and Dawson, Proc. Ro-Y. SOC. A , 1935, 148, 32. Nernst, 2. physik. Chem., 1904, 47, 52. a Miers, Proc. Roy. SOC. A , 1903, 71, 439 ; Phil. Trans. Roy. SOC., 1903, 202, 459. * Berthoud, J . Chim. Physique, 1912, 10, 624. ti Valeton, 2. Krist., 1924.59, 135, 335. 6 Spangenberg, Neues Jahrb. Miner. A , 1928, 57, 1197.C. W. BUNN Q! Minuks. I33 of the driving force for deposition. (Crystal faces of different types would be characterized by different rates of growth at the same supersaturation.) The conception seemed reasonable, but there was at that period no experi- mental evidence to support it, and in 1932 we set out to measure simul- taneously the rates of growth of crystal faces, the supersaturation at the faces, and the concentration gradients round the crystal. Miers' method would be difficult to adapt to the exploration of any variation of supersaturation at different points on the same face, and could not give concentration gradients in the solution ; a different method was therefore developed. Crystal growth was reduced practically to a two-dimensional process by growing a crystal in a thin film of solution confined between glass plates, and concentration differences were determined from refractive index differ- ences measured by an interference method 6X A which was suggested by T.R. - Scott, then of this laboratory : the glass plates were half-silvered and not quite parallel, and under the microscope in parallel monochromatic trans- mitted light, a system of interference fringes was seen, the distortions of which were a measure of changes of refractive index and thus of concentra- tion. Photographs were taken at intervals, and subsequently measured in detail. Sodium chlorate was chosen as a convenient substance for the experiments, as it is cubic, and strongly supersaturated solutions can be obtained in which unwanted additional crystal nuclei are less readily formed than in solutions of many other substances.I shall not give further experi- mental details, as they are adequately covered in a paper by Berg,' who took up this method a little later. The results of our earliest experiments will not be described in detail, as they are substantially the same as those of Berg. It will suffice to state that the supersaturation was found to vary along any one face, being greatest at the edges and least at the centre ; this banished the prospect of discovering Berg, Proc. Roy. SOC. A , 1938, 164, 79.I34 CRYSTAL GROWTH FROM SOLUTION any absolute correlation between rate of growth and supersaturation, since faces remained flat in spite of considerable differences of supersaturation at different points.Moreover, the four observed faces of a crystal, though crystallographically equivalent, usually grew at different rates, and no correlation could be found between rate of growth and either the maximum, or the minimum, or the average supersaturation a t the face. There was no question of exhaustion of solute at certain faces; in fact, often the most slowly growing faces were in contact with the most strongly supersaturated solution, My purpose now is to reconsider these facts in relation to the phenomenon of layer formation described in the preceding paper, and to describe some experiments in which for various reasons crystal growth was very abnormal ; these abnormal experiments are in some ways more instruc- tive than the more normal ones.d R D I I FIG. 2.-Supersaturation along faces, and rates of growth. Expt. I, a t 13 min. and 36 min. Initial supersaturation of solution, 7-77 g./Ioo g . solution. Experimental Three experiments will be described, which were the most remarkable and instructive of those carried out ; they were all done in 1932. In the first, the principal point of interest is in the fact that the rates of growth of the four observed faces (see Fig. I) did not slowly and steadily diminish with time as in some other experiments (see Expt. 2 below) ; the rate of growth of one face (D) diminished rapidly, within about 20 min., to half its initial value, while those of the other three faces first decreased and then increased again.The increase was most marked for faces A and B which were adjacent to D. Face A, which was at first the slowest, became the most rapidly growing. This was certainly not due to exhaustion of the solution near face D, for in this experiment the initial solution was exceptionally strongly supersaturated (7-77 g. NaClO, per IOO g . solution), and the concentration a t the crystal faces was everywhere far aboveFIG. 3.-Crystal of sodium chlorate (the cracked crystal of Expt. 2 ) growing in thin layer of solution between nearly parallel half-silvered mirrors, illuminated by parallel monochromatic light. To face page 1351C. W. BUNN =35 saturation for the whole duration of the experiment. The variation of super- saturation along the faces, shown in Fig.2, was of the same type as in our earlier experiments and those of Berg, and there was again no correlation between rate of growth of a face and the maximum or minimum or average supersatura- tion. The changes in rate of growth of the faces give the impression that the total amount of solute reaching the crystal was limited but was redistributed during the course of the experiment. Faces A, B and C in this experiment were not entirely flat ; face €3 was very slightly convex throughout, while A and C at different times exhibited temporary steps, when one half of a face gained on the other ; but these steps were soon eliminated (they are ignored in Fig. I, which gives the average rate of growth for each face). This demonstrates that in normal circumstances there is an influence (presumably the layer-spreading process) which tends to keep a crystal face nearly flat during growth, and is even able to overcome an appreciable temporary departure from flatness.Mi'rruhs. /O 20 30 40 50 60 I I I I I I FIG. +-Rates of growth of faces of cracked crystal, Expt. 2. The course of events in Expt. z is very relevant to this same point. In starting the experiment (probably when the upper mirror was put in place) the crystal became cracked right across ; each of the two opposite faces which were cracked behaved as two independent faces, the two halves growing at considerably different rates. One photograph is reproduced in Fig. 3 to illustrate the sort of experimental material on which these results are based. Fig. 4 shows the rates of growth, which in this experiment show a steady decline with time, and Fig.5 shows the supersaturation along all the faces a t two stages (there was very little change in this respect in the duration (67 min.) of the experiment). The independent growth of the two halves of a cracked face underlines what was said in discussing Expt. I ; presumably the layers were not able to bridge the crack, and each half of the face was therefore free to respond independently to whatever changes occurred either on its surface or in the solution in contact with it. As in all the experi- ments, some initial dissolution of the seed crystal occurred when warm solution came jn contact with it ; but in this case one corner was dissolved more than the others, and by the time the assemblage reached laboratory temperature, In Expt.3 the course of events was still more abnormal.136 CRYSTAL GROWTH FROM SOLUTION the crystal had a small (110) face on one corner in addition to the usual four IOO (cube) faces. This (110) face grew faster than any of the cube faces, and soon eliminated itself, its rate of growth increasing towards the end. The rates of growth of the cube faces showed the remarkable changes recorded in Fig. 6 ; face A soon stopped growing altogether] while at the same time the rate of growth of B very much increased. Face A did not grow at all for a whole hour afterwards ; the changes in the rates of growth of the others were somewhat similar to those which occurred in Expt. I. The supersaturation along the faces a t two points of time in the crucial period (at 8 min.and 13 min.) is shown in Fig. 7 and 9, and the concentration distribution round the crystal a t the same times in Fig. 8 and 10. FIG. 5.-Supersaturation along faces, and rates of growth. Expt. 2, at 32 min. Initial supersaturation of solution, 4.9 g./Ioo g . solution. FIG. 6.-Rates of growth of faces, Expt. 3. The course of events in this experiment forms the most striking demonstration of the lack of correlation between the rate of growth and supersaturation, for face A was in contact with strongly supersaturated solution, and yet stopped growing ; in fact, by the time i t had stopped growing, the average concentration of solution in contact with it was greater than for any of the other three faces The diagrams of distribution of concentration (Fig.8 and 10) suggest that, (Fig. 9).C. W. BUNN I37 FIG. 7.-Supersaturation along faces, and rates of growth. Expt. 3, a t 8 min. Initial supersaturation of solution, 3.36 g./Ioo g. I I 5 /b x /O-=cm. FIG. S.-Equal-concentrstion contours. Expt. 3, a t 8 min. Contours drawn a t intervals of 0.246 g./Ioo g. Dishme d o n 3 faces FIG. 9.-Supersaturation along faces, and rates of growth. E" Expt. 3, a t 13 min.138 CRYSTAL GROWTH FROM SOLUTION at first, the avidity of the rapidly growing (I 10) face caused an extensive reorgan- ization of the concentration distribution round the crystal ; near the (110) face, not only were the gradients steepened, but also the supersaturation near the crystal was much reduced, and the effect was to shift the diffusion centre (the region towards which solute particles were diffusing) away from the centre of the crystal and towards the (110) corner face ; it may be that this is the reason why, a t this period, faces C and D which were adjacent to the (110) face were growing faster than A and %solute was diverted from A and B towards C and D, as indicated by the arrows in Fig.8. If this were all, we should expect faces A and B to continue to grow a t about the same speed ; but in fact A stopped growing altogether while B's rate of growth increased until it was about the same as that of C. These changes are similar to, but more extreme than, those which occurred in Expt. I ; and they raise in the most acute manner the central question : what is it that determines the speed a t which a cube face of sodium chlorate grows? It is evident that the supersaturation a t the face is not the determining factor.Are the extreme variations due to some other condition in the solution, or to changes on the surface of the crystal ? FIG. 10.-Equal-concentration contours. Expt. 3, a t I 3 min. Contours intervals of 0.246 g./Ioo g. drawn at Dr. S. P. Humphreys-Owen,8 who has recently been studying the growth of sodium chlorate crystals by the same method, has observed phenomena similar to those described here-extreme variations of the rates of growth of the four observed faces, including the complete stoppage of growth of one or more faces. He has also observed the restarting of growth of a stopped face. All these phenomena must be considered in any attempt to frame a theory of crystal growth from solution.Berg, ' in considering the fact that crystallographically equivalent faces grew, in his experiments, at different rates, was inclined to attribute the differ- ences to the presence of local traces of impurities. This suggestion does not seem acceptable. Dissolved impurities would affect all faces equally. Undissolved impurities might become localized ; but it is difficult to see how they could cause the effects observed. Impurities may or may not be built into the crystal ; if they are built in, the poisoning effect would be temporary, because fresh material, deposited on the impurity, would create a fresh surface free from impurity ; if they are not built in, they remain near the surface, rejected by each layer but able to hinder growth by their continued presence ; but i t is difficult to Humphreys-owen, Proc.Roy. SOG. A (in press).C . W. BUNN I39 believe that the effects would be confined to one particular face for any length of time-neighbouring faces would be affected. Moreover, how are we to explain the fact that the rates of growth of A, B and C in Expt. I increased while that of D decreased ? It is unlikely that impurity would be built in and covered over on three faces but not on the fourth. I t is equally unlikely that impurity would migrate towards D and away from the three other faces. If we reject explanations based on local surface contamination, we must consider whether the phenomena can be explained by any other solution condition, or by surface changes arising from within the crystal (such as strains or cracks).On solution conditions, we may observe that the only condition other than supersaturation a t the face (which is clearly not the controlling factor) is the concentration gradient at the face. Concentration gradients arise in the first place because the crystal, so to speak, “ takes ” solute out of the solution and makes its own gradient ; but suppose the gradient is disturbed by convection currents : will this affect the rate of growth of the crystal face, irrespective of the supersaturation a t the face ? Suppose the concentration at the face rises and the gradient is partly levelled : will the rate of growth decrease ? I t need not : the crystal face might continue to “take” solute from the solution at the former rate, and this would restore the gradient to its former value.But the sequence of events in Expt. 3 suggests that a local steepening of gradient due to the rapid growth of a small (110) face led to an increased rate of growth of the neighbouring cube faces; and this suggests that changes of gradient due to external causes such as convection currents might similarly affect the rates of growth of cube faces. The existence of slow currents (due to variation of density of the solution and to heat of crystallization) was confirmed in one experiment by watching the movement of colloidal particles deliberately put into the solution ; such movements might take unsymmetrical paths and thus affect different cube faces unequally.We are thus led t o enquire whether there is any justification for regarding diffusive flow of solute as not merely an effect following on surface reactions controlled by other factors, but on the contrary as a process which (once it is started) has a positive influence of its own. I n discussing this question, possible relations between the phenomena of layer formation and the conditions in the solution will be considered. Discussion Surface Profile in relation to Supersaturation at the Face. The fact that the supersaturation of the solution at a crystal face varies all along the face shows that the possibility discussed by earlier theorists that the rate of growth (G, the thickness * of solute deposited on a face in unit time) is some function of the supersaturation S at the face is not correct in its simplest form, since a crystal face usually remains substantially flat in spite of considerable differences of supersaturation at different points.But this theoretical possibility was based on the assumption that a crystal face is a uniform surface. I t has, however, been pointed out in Part I that a crystal face may be effectively a high-index surface at the centre and predominantly a low-index surface near the edges ; consequently the surface forces vary from the centre to the edge of a face, and if one postulates a relation it must be with the proviso that K may vary all along any one face; it would be high at the centre of a face where there is a high-index surface, and low towards the edges where there is predominantly a low-index surface.I t may be that supersaturation and surface character are mutually adjusted so that for any one face at a particular time Kf(S) is constant. Furthermore, the lack of correlation between the rates of growth of different faces and G = Kf(S), * G represents the average thickness deposited all over the face, ignoring the fine structure of the surface.CRYSTAL GROWTH FROM SOLUTION the supersaturation might be due to differences of surface profile which completely mask the influence of supersaturation. The experimental evidence either means this, or else it means that the magnitude of the supersaturation at the face plays no part in determining the rate of growth (apart from the basic fact that no growth occurs unless the concentration at the face exceeds the saturation value).There appears to be little hope of securing experimental evidence on this question, for it is difficult to see how to discover the crystallographic charac- ter of layer edges while growing. But even if it were demonstrated that there is a definite relation between the character of the surface, the supersaturation and the rate of deposition, it would still leave the main problem unsolved : at a given supersaturation the rate of growth might have a wide range of values depending on the surface profile which is set up during growth. I t is necessary to enquire what determines the type of surface which is set up. It is likely that the key to the situation is to be found at the centre of a crystal face, where layers normally originate. It is a striking fact that normally layers spread from the centres of crystal faces, where the supersaturation is lowest.(It is assumed here that the findings on sodium chlorate are typical. A few preliminary experiments on one other substance, potassium ferricyanide, showed that for this crystal also the supersaturation is lowest at the centres of the faces. Further work on other substances is desirable.) The magnitude of the supersaturation evidently does not control the inception of layers ; and this is consistent with the facts about the rate of growth of faces. The other solution conditions which may vary over a crystal face are the concentration gradients which control the diffusive flow of solute ; and we therefore turn to a consideration of the diffusion field. According to Berg, more solute arrives at the centre of a face than at the edges, and since the face remains flat, the excess must be dissipated by surface migration.While it may be doubted whether the excess is as large as the 25-50 yo stated by Berg (the accuracy of measurement of gradients is not great enough to give confidence in the magnitude, which would mean an enormous surface migration), nevertheless, if we may accept the indication of an excess at the centre rather than at the edges, this suggests an obvious explanation of the inception of layers at the centres of faces. Moreover, the arrival of excess solute at the face centre can hardly be attributed to events on the crystal surface but must be due to the diffusion process. This line of thought leads us to ask whether radial inward diffusion to a polyhedral crystal necessarily tends to deliver excess o€ solute to the centres of faces.If there were no surface migration, the crystal surface would " take" from the solution uniform amounts of solute all along the face, and this would impose on the diffusion field a particular concentration distribution capable of supplying uniform amounts of solute to the surface. But if surface migration may occur, even if only to a small extent, the diffusion process is not tied down to delivering uniform amounts of solute, since any non- uniformity can be dissipated by surface migration. In these conditions, radial inward diffusion to a polygonal crystal plate is unlikely to deliver uniform amounts of solute along each face : the amount arriving at the corner is likely to be different from that arriving at the centre.Precise mathematical treatment of this problem does not appear possible, but by the following approximate numerical procedure (suggested to the writer by Sir Cyril Hinshelwood) it is possible to draw some instructive conclusions. Round a square representing the crystal plate, the field is divided into small squares, each containing a number representing supersaturation. At first the numbers are all equal; to represent the start of crystal growth, Layer Formation in relation to the Diffusion Field.C. W. BUNN 141 the numbers in the squares next to the crystal are all reduced by the same amount, representing the amount of solute taken out of the solution by the crystal in unit time. The diffusion process is represented by calculating the change in each figure in unit time due to transfer of solute to or from each of the neighbouring squares, using an arbitrary diffusion constant.Alternate stages of deposition and diffusion are then carried out ; in doing this, various assumptions about deposition can be made-for instance, uniform deposition along the face a t a rate which can be either constant or diminishing with time (these assumptions corresponding to absence of surface migration), or deposition from each square at the rate at which solute arrives by diffusion (this corresponding to the assumption of surface migration which dissipates any non-uniformity of arrival). The diffusion field spreads outwards with each successive stage, just as it does in practice.When uniform deposition was enforced at a rate slowly declining with time (to imitate the experimental conditions), this led to the establishment of a diffusion field similar to that found experimentally (in normal undisturbed circumstances), with equal- concentration contours which in the outer region are nearly circular, but near a 6 FIG. I I a.-Form of equal-concentration contours in symmetrical growth, obtained by b. If, at face A, a decline of gradient and rise of concentration calculation. occurs, this leads to a greater convergence of solute on B and D. the crystal have a shape which is a compromise between square and circular (as in Fig. 11 a). The amount of solute arriving at each square along the face was nearly uniform, but there was a persistent small deficiency at the corner : thus, at each diffusion stage, slightly too little solute arrived near the corner, but at each deposition stage, uniform amounts were taken out of each square.The procedure was then altered, to study the tendency of the diffusion process when not tied down to uniform delivery at the face ; what- ever solute arrived at each point was assumed deposited there. This had the effect of making the deficiency of arrival at the corner more and more marked at each successive stage, and the inner concentration contours became gradually more nearly circular. This procedure is slightly unrealistic, in that the “ crystal’’ does not actually grow ; and the operations are approximate ; but it does represent the essentials of the present discussion, and the effect appears to be a genuine indication of the tendency of the radial diffusion process to deliver less solute to the corners than to the centres of crystal faces.We may regard this as due to the tendency of the diffusion field towards the circular symmetry appropriate to radial inward diffusion : the outer contours which are nearly circular try to impose their symmetry on the inner ones ; and the more nearly circular are the inner contours, the greater the excess of solute arriving at the centre of a face. The tendency would be still more marked142 CRYSTAL GROWTH FROM SOLUTION in three-dimensional diffusion, and the excess arrival at the centres would build up as far as surface migration allows.This conclusion not only suggests the explanation of the inception of layers a t face centres which has already been stated, but it also has implica- tions on the question of changes in rates of growth; for if the normal inception of layers at face centres is due to diffusive convergence of excess solute there when the diffusion field round the crystal is undisturbed, then any disturbance of the field by convection currents is likely to change the rate of inception of layers unequally on different faces. The suggestion already made, that externally caused gradient changes may be responsible for changes in rate of growth (irrespective of supersaturation at the face), is thus supported. The simplest way of regarding the inception of layers at face centres (if the present views are correct) is the following.If the shape of the crystal depended on the diffusion process alone, it would be spherical; but in fact the crystal opposes this tendency and forms nearly flat faces. The formation of a low pyramid of growing layers represents the attempt by the diffusion process to make the crystal spherical-an attempt which is not very success- ful, since the layer system on a growing crystal face is a scarcely perceptible departure from flatness. The example quoted in Part I, in which layers spread from the corner of a lead nitrate crystal following the disappearance of a small rapidly growing (presumably high-index) face there, is consistent with these views. A rapidly growing face sets up a steep, strongly convergent diffusion field (cp.the IIO face in Expt. 3 above) ; as soon as this face disappears and is replaced by a corner, this field initiates layers on the normal faces which meet at that corner. Surface Profile in relation to Concentration Gradients. The problem of rate of deposition on a crystal surface will now be approached in a different way, in an attempt to understand more closely the relation between the character of the deposition surfaces and the solution conditions. It has been noted in Part I that deposition takes place on the edges of layers which are apparently high-index surfaces; at the centre of a crystal face the layers are sometimes so thin that this part of the surface may be regarded as a low pyramid of vicinal faces, while towards the edges of the face where the layer edges are thicker, the surfaces are often irregular-possibly irregular on the molecular scale ; in any case the term " high-index surface " covers all deposition surfaces.Further, there is a tendency for such surfaces to '' heal "-that is, for deposition to occur in such a way that the new surface has lower indices; the very formation of thick layers is the first symptom of this tendency, and its further progress is indicated by the tendency for the layer periphery to become more regular as growth slows down. The slow decline of rate of growth with time in undisturbed conditions (see Fig. 4) may be due to gradual healing. Rapid crystal growth, in fact, appears to depend on the maintenance of sensitive high-index surfaces ; if the surfaces were to heal completely, growth would be severely inhibited ; it is possible that the complete stoppage of growth of faces which has sometimes been observed is due to complete healing.The outstanding question which arises is the following : what is it that prevents complete healing in normal circumstances? If a crystal having imperfect surfaces is put into a supersaturated solution, why is it that the first solute molecules do not deposit in such a way as to make low-index surfaces which would then grow no further ? The supersaturation is certainly not the controlling factor here, because stopped faces are usually found to be in contact with the strongest solution; and it is difficult to imagine anyC. W. BUNN I43 other solution condition except the diffusive flow of molecules set up by the initially formed gradient.It may be that if solute molecules arrive fast enough, they are deposited in an indiscriminate way so that high-index surfaces are maintained. This suggestion appears to imply that when solute molecules are moving towards the crystal surface, the increased component of Brownian motion towards the surface increases the chance of deposition on sites which preserve a high-index surface. The change in the component of molecular velocity towards the surface which diffusive flow implies is, of course, small ; but the chance of deposition on such sites might increase appreciably with quite a small change in the component of motion towards the surface. The chance of deposition on a site of particular crystallographic character may depend on the proportion of molecules having velocity components towards the surface which exceed a critical value ; and the critical value may be quite delicately related to the crystallographic character of the site.If it is true that healing is prevented by a sufficiently rapid diffusive flow of molecules towards the surface, we may imagine that exaggerated effects may follow a local levelling of the gradient by convection currents : the surface partially heals, and becomes less capable of receiving solute, and this may lead to a further decline of gradient. How far such a progressive change would go cannot be predicted, but the total stoppage of growth, which sometimes occurs, indicates that it may go to extreme lengths in this direction. Further, if the rate of growth of one face decreases, the levelling of gradients there will lead to a greater convergence of diffusing solute on neighbouring faces (Fig.11 b), and consequently to an increase in their rate of growth ; the differences between the rates of growth of iieighbouring faces of the same crystal are thus still further exaggerated. (This is presum- ably part of the explanation of the course of events in Expt. I and 3.) The restarting of growth of a stopped face which has been observed by Humphreys-Owen might be explained in a similar way as being due to the external building-up of a gradient and consequent diffusive flow which leads to the formation of a layer nucleus; or alternatively this might be truly a chance phenomenon: on a perfect cube surface in contact with homo- geneous supersaturated solution, the chance of formation of a layer nucleus is very small but not zero.Conclusion. Much of the foregoing discussion has been concerned with the diffusion field and the surprisingly important part it appears to play in layer formation : the role which supersaturation might have been expected to play is in fact (according to the present interpretation) taken over by the concentration gradients. Nevertheless, the part played by diffusive flow is only one side of the picture ; indeed, chronologically it is a secondary part. When a crystal is put into a homogeneous supersaturated solution, there are a t first no gradients ; it is only when the crystal " takes " solute out of the solution that gradients are created; the " taking " of solute out of the solution by the crystal is the primary process.It is perhaps here that the magnitude of the supersaturation at the surface plays its part : the initial rate of deposition may depend on the supersaturation as well as on the nature of the surface. (The growth of a crystal nucleus spontaneously formed in a solution is a different matter ; the " taking " of solute from the solution is part of the process of nucleus formation, which will not be considered here.) The extreme importance of the nature of the surface has already been emphasized; it is possible that if a sodium chlorate crystal with perfect cube faces were put into a supersaturated solution, it would not grow at all, or at any rate the beginning of growth might be delayed an indefinite t h e .144 GROWTH OF FACES OF SODIUM CHLORATE CRYSTALS (It would be difficult to test this, because any seed crystal grown for the purpose, when taken out of its solution, is unlikely to have perfect surfaces even if it had them originally, owing to drying of mother liquor which would be likely to give irregular surfaces.) The present work gives no information on the possible connection of rate of deposition on a given surface with the magnitude of the supersaturation, because the precise crystallographic character of the deposition surfaces is not known.It does not seem possible to determine the character of the deposition surfaces in normal growth ; but it might be worth while to study growth on more extensive high-index surfaces which are deliberately created by partial dissolution. The character of the surfaces would change from the moment growth started, and it would therefore be necessary to measure the rate of growth from the earliest possible moment and extrapolate back to zero time. Only by observing growth in the earliest possible stage is i t likely that any significant relation between character of surface, supersaturation and rate of growth would emerge. The conception of crystal growth developed in this paper is not unlike the current conception of certain chain polymerization reactions depending on activated molecules, where the initiation of chain formation depends on activation (photochemically or by free radicals), and its continuation depends on the maintenance of activated chain-ends. In crystal growth, high-index surfaces appear to be the active surfaces which are capable of adding on further molecules ; the beginning of growth depends on the presence of such surfaces, and its continuation depends on their maintenance. The problems of rate of crystal growth can be divided into a study of the factors con trolling initiation (the magnitude of the supersaturation a t the surface may play a part here), and those controlling the maintenance of active surfaces (in which the rate of diffusive flow of solute appears to play an important part). Deposition on high-index surfaces appears to be the key to problems of crystal growth. I wish to thank Mr. H. Emmett for his help in the experiments and photography involved in this work, Miss B. J. Botterill for assistance with calculations, Dr. S. P. Humphreys-Owen for the opportunity of reading his paper before publication and for a stimulating discussion, and those of my colleagues who were good enough to read the paper and offer comments and criticisms. I.C.I. Ltd., Plastics Division , Welwyn Garden City, Herts.

ISSN:0366-9033    

DOI:10.1039/DF9490500132

出版商:RSC    

年代:1949

数据来源: RSC