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.