SUPERSATURATION AND CRYSTAL SIZE BY WILDER D. BANCROFW
Since von Weimarn was the first man to make a systematic study of the form in which substances precipitate from solution under varying conditions, it is necessary to take von Weimarn’s theory1 as a starting point. “The process of candensation [or precipitation] depends on a number of very different factors: on the solubility of the substance, on the latent heat of precipitation, on the concentration a t which the precipitation takes place, on the normal pressure a t the surface of the solvent, on the viscosity of the solvent, on the dielectric constant of the solvent, and on the molecular weights of the solvent and of the precipitate. It is of course impossible t o take into account now the effect of all these complex factors on the precipitation. We will therefore simplify the problem by considering only two of the factors, the solubility of the precipitating substance ‘and the concentration at which the precipitation begins, or, in other words, the initial concentration of the precipitating molecules. The process of condensation can profitably be divided into two parts. I n the first stage the molecules condense to invisible crystals or to crystals which can only just be seen in the ultra-microscope under the most favorable conditions-molecular complexes. From a molecular kinetic point of view, the first stage of the condensation process takes place exactly like a reaction in a solution. The velocity a t the important first moment of the first stage of the condensation can be formulated as follows: condensation pressure = Q -L = P = KU ( I ) ’ W = K condensation resistance L where W is the rate of precipitation in the first movement, Q is the total concentration of the substance which is to precipitate; I, the ordinary solubility o$ the substance in coarse crystals; Q-- L = P the amount of supersaturation. The “Grundziige der Dispersoidchemie,” 39
(191 I).
Sueersaturation and Crystal Size
IO1
ratio of P/L = U rhay be called the specific supersaturation [percentage supersaturation] at the moment when precipitation begins. One must not forget that this formula applies only to the first stage of the condensation process when the excess molecules and their smallest complexes come in contact in the dispersing medium and condense. It is only under these conditions that the kinetic conceptions have a meaning analogous to those which have proved useful when studying the kinetics of homogeneous systems. “The second stage in the process of precipitation is concerned with the growth of the particles as a result of diffusion. The rate of this precipitation, according.to the Nernst-Noyes theory, is given by the equation D v=-.O.(C-Z) 6
(2)
where D is the diffusion coefficient, 6 the length of the diffusion path (the thickness of the adherent film), 0 the surface, C the concentration of the surrounding solution, and 1 the solubility of the particles of the dispersed phase for a given degree of dispersity. C - I is the absolute supersaturation. * These two equations call for some comment. Direct experiment shows that, under otherwise similar conditions, the rate a t which precipitation occurs is greater the larger the value of P and that for the same value of P the rate of precipitation is greater the smaller L is. It follows experimentally, therefore, that the rate of precipitation varies directly with U = P/L increasing or decreasing as U becomes larger or smaller. The experiments do not prove that the rate of precipitation is directly proportional to U. The reason for assuming such a proportionality in Equation I is t h a t hat is the simplest, most natural, and, theoretically, most pobable relation as well as the one which is the easiest to apply in an analysis of the Equation z is not only the result of the theory of Noyes and Nernst,. but has been verified brilliantly by Johann Andrejew in an extensive series of experiments on the rate of growth of crystals of citric acid, orthochrodinitrobenzene (in ether), and sodium chlorate. Jour. russ. chem. Ges., 40, 397;. Zeit. Kolloidchemie, 2, 236 (1908).
1
W i l d e r D. Bavcroft
102
phenomenon of precipitation. The more characteristic cases of precipitation become quite clear when considered with reference to Equations I and 2 , and one is able to classify many phenomena of condensation which .at first sight seem quite unrelated., “The equation V = D . 0 (C - Z)/& shows that the growth of the particles of the dispersed phase is cut down very much in two cases, when either C - 1 or D is very small. The first occurs in very dilute solutions of the reacting substances, and the second in very concentrated solutions when the great rate of precipitation W in the first moment causes the formation almost instantaneously at many different points in the dispersing medium of particles which are not molecularly dispersed and which do not diffuse rapidly. These particles agglomerate but slowly. . . . . . . . . . , . . . . . . . . . “By means of Equation I , W = K (condensation pressure/ condensation resistance) = KP/L = KU, the following phenomena among others may be easily explained and made to seem matters of course. With a condensation pressure of several grams (P) sparingly soluble substances such as A1203.3H20 and AgCl are obtained in the form of a very fine jelly or of a curdy precipitate due to the coalescing of very fine crystals, while readily soluble substances such as NaCl are obtained as small crystals. From Equation I we see that in the first case the condensation resistance was small and therefore the rate of precipitation of W very large, while the rate of precipitation is very small in the second case because the resistance to condensation is large. Prom Equation I , however, we see what conditions must prevail in order to reverse the results previously obtained with Al2O3.3H2Oand NaC1. If we carry out the formation of A1203.H20in strongly ammoniacal solution in which this substance is a good deal more soluble than in water, we shall get microcrystals. If we precipitate sodium chloride by letting HC1 and NaSCN, or still better sodium alcoholate,l react in a mixture of ether and amyl alcohol in which NaCl is practically insoluble, we 1
Jour, russ. chern. Ges., 40, 1127 (1908).
~
Supersaturation and Crystal Size
103
get a cirdy precipitate very like that of AgC1. These illustrations when taken in connection with Equation I are very convincing. Any reasoning man will see that there is no such thing as a specifically amorphous colloidal substance and that one can obtain any substance in large crystals, no matter how insoluble i t may be, by making P and consequently W very small. I n extreme cases, when I, is vanishingly small, it will of course take seas of reacting solutions and centuries of time to produce large crystals; but it would be just as foolish to doubt the possibility of getting large crystals as it would be to say that NaCl does not occur in crystalline form because it can be obtained in a curdy state. “We must now take into account that the terms ‘practically insoluble’ and ‘practically instantaneous reaction or precipitation’ are very flexible. I n most cases the rates of precipitation, W, of practically insoluble substances from reactions between solutions cannot be differentiated quantitatively by our senses and appear to be equally instantaneous for ‘practically insoluble’ substances which, however, have different solubilities. We cannot doubt, however, that some of these rates are really ten-, a hundred- or a thousand-fold some of the others, just. as the practically insoluble substances vary very much in solubility. While we cannot of course measure the true reaction velocity, W, in many cases, we can judge its change from the value U which is proportional to it. To bring this out clearly we give a table for U and. W a t constant P but varying I,: P 10-2
-
U
IO-^
I02
10-2
10-5
103
10-2
10-6
104
10-2 10-2 10-2
10-7
105
10-8
I06
IO-^
IO’
Io-2
10-10
I08
W
K .102 K . IO$
K.1 0 4 K.10.j K .IO^ K.1 0 7 K.IO*
Wilder D. Bancroft
104
“From this table it can be seen that with equal P b u t varying U, such as U = 102 and U = IO^, we cannot get dispersed systems of identical properties because the rates of precipitation vary markedly, from W = K . 1 0 2 to W = K . 1 0 ~ .It must be noted that for an understanding of the formation of suspensoid solutions it is very important to know that the specific concentration C/I is the important thing and not the absolute concentration C. It would be entirely false, however, to say that the absolute concentration is unimportant, for the latter determines whether a jelly or a suspensoid solution is formed. An equal and high value for W can be obtained in two ways, either by increasing P when I, is fairly large or by decreasing P when I, is very small. I n the second case the dispersed particles are far distant one from another and we get a suspensoid solution or suspension, and in the first case we get a curdy or gelatinous precipitate.’’ Von Weimarn2 considers that with increasing supersaturation five stages can be distinguished fairly definitely in the case of sparingly soluble substances, though there is, of course, no sharp dividing line between the stages. For slight supersaturations, no precipitation occurs inside of several years. This may be called the stage of colloidal solutions. I n the next stage of higher supersaturation perfect crystals are obtained in a relatively short time. In the third stage skeleton crystals and needles are obtained. When the fourth stage is reached, a curdy, apparently amorphous, precipitate is obtained, and in the highest stage of supersaturation a jelly is obtained. It is immaterial whether all these precipitates are crystalline or not. Von Weimarn considers them all crystalline; but he considers a liquid and a gas as crystalline. This may be true, but it is not helpful because we should then have to invent another word to describe what most people mean by crystalline and that would be confusing. It is undoubtedly true that many precipitates are called amorphous when they are really crystalline; but if a liquid is 2
Zeit. Kolloidchemie, 5, 157 (1909). “Zur Lehre von den Zustanden der Materie,”
IO
(1914).
Supersaturation. and Crystal Size
10.5
amorphous, a precipitate consisting of supercooled and very viscous drops is also amorphous. Von Weimarnl has tested his theory on barium sulphate, among other substances, by mixing equivalent solutions of MnS04 and Ba(SCN)2. When the reacting concentrations were approximately N/20000 - N/7000 no precipitate was formed in the course of several years. With concentrations of about N/7000 - N/6oo perfect crystals were obtained, the rate of crystallization being very small with the more dilute solutions. With initial concentrations of about N/600 to 0.75 N skeleton crystals and needles are formed. When the initial concentrations are about 0 . 7 5 N - 3 N the precipitate is curdy, becoming gelatinous at the higher concentrations, while jellies are obtained, at least temporarily, when the concentrations are about 3 N - 7 N. Satisfactory results have been obtained by von Weimarn with an extraordinary number of salts and he claims never to have found an exception. I n spite of that the general presentation does not seem to me ideal. While it is apparently true that one can get any salt coming down in a gelatinous form if the concentrations of the reacting solutions are sufficiently high, those are not the conditions under which jellies are usually obtained. Then von Weimarn’s conditions hold only for the case where the solutions are not stirred after mixing. He separates the colloidal solutions from the curdy precipitates in spite of the fact that one can often pass directly from one to the other. The effect of a slight excess of one of the reacting substances is not discussed effectively, though the matter receives a perfunctory and formal consideration in the later work,2 where the effect of viscosity is also discussed. I have found von Weimarn’s views distinctly more helpful if one discards the formulas and restates the whole thing from another point of view. The mean size of the crystals is determined by the total amount of material crystallizing and the number of crystals. The really important thing there-
’ “Zur Lehre von den Zustanden der Materie,” 2 1 (1914). Von Weimarn: Ibid., 6 (1914).
~
106
N'ilder D . Bancrqft
*
fore is the number of nuclei which are formed under any given conditions and von ,Weimarn appears to have overlooked this factor entirely. T h e difference which this may introduce can be seen very clearly from the behavior of very dilute solutions. Von Weimarn has deduced very properly that the condition ' favorable to the growth of large crystals is the existence of a very slight supersaturation, but he does not lay stress on the fact that the reason for this is that there is then only a very slight tendency for the solution to crystallize spontaneously and consequently the crystallization takes place only on the crystal which is to grow. If one stirs such a solution vigorously, a number of nuclei are formed and we get colloidal solutions because the number of nuclei is so large that the excess material comes out chiefly as nuclei. There has been no direct experimental study, so far as I know, of the way in which the number of nuclei varies with the increase in supersaturation; but we can deduce i t empirically from the observed behavior of supersaturated solutions. For a definite rate of stirring the number of nuclei seems to increase a t first less rapidly than the supersaturation and afterwards more rapidly. This means that we get colloidal solutions of lead chromate, for instance, if we mix very dilute solutions of potassium chromate and lead nitrate,' because the number of crystals is so great that the total mass of each one is very small. At a little higher concentration we get flocculent or curdy precipitates because of the agglomeration of the crystals. At still higher concentrations, the number of nuclei has apparently not increased proportionally, for the crystals are distinctly larger. With continuously increasing supersaturations the crystals pass through a maximum size and then begin to decrease either because the number of nuclei are increasing more rapidly than the concentration or because the rate of crystallization becomes so high that good crystals are not formed. At very high degrees of supersaturation, gelatinous precipitates are formed. This was known before in isolated cases, as for 1Free: Jour. Phys. Chem., 13, 114 (1909).
Supersaturation and Crystal Size
107
instance when concentrated sodium carbonate and calcium chloride solutions are mixed ; but von Weimarn was apparently the first to formulate the general statement. I am not certain whether a gelatinous precipitate is a limiting case when the size of the crystals becomes very small. It seems to me quite possible that we are dealing with the appearance of a second liquid phase; but I shall probably discuss this in a later paper. Empirically we find that the relative number of nuclei decreases with rising temperature because we get more coarsely crystalline precipitates at higher temperatures, a point which is apparently not covered by von Weimarn’s theory. The question of adsorption receives very little attention from von Weimarnl in spite of the fact that barium sulphate is well known to precipitate more coarsely crystalline in presence of sulphuric acid or hydrochloric acid and very finely divided in presence of barium chloride, This has been discussed in detail by Weiser2 so I need not go into it. The general results of this paper are as follows: I . The theory of von Weimarn as to the relation between degree of supersaturation and size of crystal is inadequate because it does not take into account the number of nuclei formed. 2 . With vigorous stirring and increasing supersaturation we get : colloidal solutions, curdy precipitates, fine crystals, coarser crystals, fine crystals, gelatinous precipitates in the case of a salt which normally precipitates anhydrous. 3. Precipitates are normally more coarsely crystalline at higher temperatures than a t lower ones. 4. Von Weimarn’s theory applies explicitly to solutions which are not stirred. The conditions for forming large crystals are those in which there is no spontaneous formation of nuclei and in which the rate of crystallization is so slow that branched crystals do not form. 5 . The effect of adsorption on the growth of crystals is not covered adequately by von Weimarn. Cornell University “Grundziige der Dispersoidchernie,” 59 (191I ) . Jour. Phys. Chem., 21, 314 (1917).
