Kinetics and mechanisms of precipitations

ChemicolEngineering Science, 1972,Vol.

27, pp. 1293-1313.

Pepamon

Press.

PrintedinCh%

Britain

Kinetics and mechanisms of precipitations D. J. GUNN and M. S. MURTHYt Department of Chemical Engineering, University College of Swansea, Swansea, Wales (First received

6 July 197 1; accepted21 October

197

1)

Abstract-The

kinetics of crystal growth of a number of sparingly soluble salts have been studied. The crystallization of barium sulphate was studied in detail; the concentration of the crystallizing solution and the surface area of the precipitate were measured as the crystallization proceeded, and the particle size distribution was measured at the termination of the growth period. The induction period was measured for three sparingly soluble salts at a number of different concentrations. The rate of growth of barium sulphate was found to show a third order dependence on concentration when the crystals were small. For larger crystals at low supersaturations the dependence was found to be first order. The measurements of concentration, surface area and particle size distribution were used to calculate the rates of primary and secondary nucleation of barium sulphate. Other evidence concerning secondary nucleation is presented and found to be consistent with the kinetic measurements. The development of crystal shape is related to diffusion in the solution and the surface reaction. It is shown that the induction period may be simply related to the rate of growth of small crystals. INTRODUCTION

ONE OF the fundamental difficulties of synthesis in the study of crystallization is to resolve the very great mass of conflicting experimental data and theories on the kinetics of nucleation and growth. This is particularly true of studies on the crystallization of salts of small solubility where information is more than usually conflicting. The difficulty can be seen in studies of the rate of growth of barium sulphate. Christiansen and NielsenHI have studied the nucleation of barium sulphate and of silver chromate using a rapid mixing technique in which two liquids containing the precipitating ions were passed into a mixing junction leading to a flow tube. The position of the earliest point of formation of visible crystals in the flow tube was used to infer the time interval between mixing and growth, and the dependence of the time interval upon the concentration of precipitants was studied. The time interval was presumed to correspond to the precipitation of a certain fixed fraction of solute. They suggested that the rate limiting step in the formation of a nucleus corresponded to the transition of a

crystal embryo containing seven ions to one containing eight. La Mer[2] showed that the results of Dinegar and La Mer could also be interpreted according to this mechanism, but that the combined results of Christiansen and Nielsen, and Dinegar and La Mer suggested that the crystal embryo contained six ions. Tumbull[3] examined the rate of precipitation of barium sulphate by measurements of electrical conductivity. He concluded that all but a negligible number of nuclei were formed during the mixing process; the number of nuclei apparently depended upon the relative concentration of the precipitants, barium nitrate and potassium sulphate. Walton and Hlabse[4] used a light scattering technique to investigate the same precipitation and concluded that the number of nuclei was independent of the concentration of precipitants. On the basis of this finding they suggested that nucleation was heterogeneous and took place on impurity particles in the solution. Nielsen[5] has reported a variable power law dependence of the velocity of nucleation of

tPresent address: Department of Chemical Engineering, Indian Institute of Science, Bangalore, India.

1293 CES Vd.

27 No. 6-G

D. J. GUNN

and M. S. MURTHY

barium sulphate upon concentration. For precipitant concentrations greater than O-01 g/l., his experimental results supported an assumption of spontaneous nucleation where the rate was approximately proportional to the eighteenth power of concentration. For smaller concentrations the rate was approximately proportional to concentration to the fourth power. There is also uncertainty concerning the interpretation of measurements of the rate of growth. It has been shown[61 that the measurements of Tumbull[3] suggest a dependence of growth rate upon ionic supersaturation to the third power, but another analysis of the same experimental results[51 supports a fourth order power dependence. O’Rourke and Johnson[7] have analysed conductimetric measurements of barium sulphate solutions during precipitation, and also find a fourth order dependence of the rate of growth upon supersaturation, but Nancollas and Purdie[8] found second order dependence for growth in seeded solutions, while Collins and Leineweber[9] report agreement of their experimental data with an equation based upon first order growth. These examples concerning barium sulphate give some idea of the difficulty of unequivocal interpretation, although the difficulty has been increased because in most cases only the concentration of the solution has been measured. Other quantities of physical importance such as the surface area available for growth and the particle size distribution were not measured. The observations on barium sulphate are similar in kind to those made on the crystallization of a number of other salts of small solubility

EXPERIMENTAL

PROCEDURES

(a) Measurement of induction period When a solution of the precipitant is mixed and examined by a beam of light passing through the solution a number of light scintillations may be seen after some time. The time interval between mixing and the appearance of crystals is known as the induction period. The equipment used in measuring the induction period is shown in Fig. 1. One of the reagents was placed into a glass beaker inside a temperature-controlled, perspex observation chamber; the second reagent was placed into a small glass tank also enclosed in the observation chamber, and both reagents were allowed to attain the temperature of the chamber. When both reagents had reached the controlled temperature, a solenoid valve was operated and the second reagent was discharged into the vessel holding the first. The magnetic stirrer was rotated at a speed sufficient to give good mixing within 2 set, but not high enough to cause heavy vortexing. A collimated light beam illuminated the vessel.

Thermosto

Heater

[lOI. In this paper, a series of experiments on the precipitation of barium sulphate, magnesium hydroxide and magnesium ammonium phosphate are reported. Of the three salts, barium sulphate was studied in the greatest detail; the observations included measurements of concentration, surface area, particle size distribution and a study of the dependence of induction period upon the initial concentration of precipitant.

Col~i,h~d beam

Fig. 1. Experimental

1294

arrangement for measurement induction period.

of the

Kinetics and mechanisms of precipitations

interval of time small scintillations of light were seen to appear in the light beam and the time of first appearance was recorded. Experiments of this type were carried out for different concentrations of reagent at a number of temperatures in the precipitation of barium sulphate, magnesium hydroxide and magnesium ammonium phosphate, all salts of small solubility. After

an

(b) Measurement of particle surface area and size distribution The instrument used for this purpose was a Bound-Brook photo-sedimentometer. The reagents were added to the observation tank of the photo-sedimentometer. A narrow beam of light passed through the rectangular section of the observation tank and the intensity of the beam was measured by a photo-electric cell and recorded on a strip chart. The liquid in the tank was agitated by a mechanical paddle at periodic intervals so that the particles were maintained in suspension. The intensity of transmitted light fell with time, and agitation was continued until the intensity became constant with time. This was taken as an indication that crystal growth had ceased; agitation was stopped and the particles then sedimented under gravity. The particle size distribution was calculated from the strip chart recording, in which it is necessary to relate the par$cle size distribution of the sample to the intensity of transmitted light and the time of observation. In all cases the volumetric concentration of solid was less than 0405 per cent so that the disturbance of the drag force acting on the particles due to other particles could be neglected[l 11; therefore the particles in fall obeyed a law similar to that of Stokes’ for spheres. The application of a photosedimentometer to the measurement of a particle size distribution of cyclonite has been considered by Bransom, Dunning and Millard [ 121, so that only a brief description is necessary here. The method of measuring the particle size distribution at the end of the crystallizing period will be described with reference to Fig. 2.

I

L

c

-I I_

I

-

H Incident

light

Susgeion

Fig. 2. Diagram of suspension cell in the photosedimentometer.

Let n(r) be the number of particles in unit volume of size greater than r, where a particle of size r has a projected area of ?rP, and a true surface area of a&. The intensity Z of light transmitted by the solution is related by Beer’s Law to the projected surface area A, of particles in unit volume of suspension, to the length L of light path in the suspension, and to the intensity I0 of light transmitted by the clear liquid. The law has the form: In ‘ip- =A&. 0

(1)

The equation is valid for dilute suspensions and may be used to calculate the surface area of crystals in the suspension during crystallization or during sedimentation. During the period of sedimentation there is no agitation. The particles sediment at a terminal velocity I/ that is given by the condition in which the drag force f acting upon the particle is equal to the weight of the particles in the liquid; f =

6mrVcLp = : ?rrQ, - p)gy

(2)

where /3 is the ratio of the drag force acting upon a particle of size r to the drag force acting upon a sphere of radius r moving at the same velocity, and y is the ratio of the volume of a particle of size r to the volume of a sphere of radius r. No change in absorption is found until the

1295

D. J. GUNN

and M. S. MURTHY

largest particles in the suspension have had sufficient time to fall a distance H from the free surface of the suspending medium to a point just below the light beam. The last particle of radius r(t) to pass out of the illuminated zone is found from Eq. (2) on substitution of H/t for V r(t) =

J(

4.5 /.LHP

t(p, - PkY

(3)

*

The projected area at time t after the onset of sedimentation is related to n(r) by the equation, A,=m

r(t) J0

dn(r) ---;i;-Pdr=iln

(4)

If both sides of this equation are differentiated with respect to t we find that,

The shape and surface structure

dn(r) --=Id dr

TLP dt

where the variable r now corresponds to the time t under consideration. An expression for dtldr is obtained from Eq. (3). The most useful equation gives - dn(r)ldr as a function of r. It is: ---dn(r) dr

from tables of standard equivalent conductances, and also with an expression given by Onsager [ 131 that takes into account the reduction due to the Debye-Hiickel cloud. The comparison of the three relationships is shown in Fig. 3. The agreement of the experimental values of conductivity with values of conductivity calculated from the initial concentrations of the precipitating solutions and standard equivalent conductances is close but not complete. However, the degree of accuracy suggested by the comparison shows that by measuring conductivity of the solution during crystallization, the changes of concentration can be followed with good relative accuracy.

9/.~H/3 - - vLr%,

- pkr

.

(6)

700

This equation may be solved from the measured dependence of lo/Z upon t, Eq. (3), and estimates of the factors /3 and y. (c) The measurement

600

of concentration

For the precipitation of barium sulphate, a conductivity cell and a Wayne-Kerr bridge were used to measure the conductance of the solution during crystallization. Standard solutions of barium chloride and sulphuric acid were added to a freshly cleaned conductance cell fitted with bright platinum electrodes. The conductivity was measured at intervals of time by means of the bridge and the concentration of barium sulphate was found from the conductivity measurements by calibration of the cell. The relationship between conductivity and concentration of the precipitating solutions obtained by calibration was compared with the conductivity-concentration relation obtained

of the crystals

Crystals of barium sulphate, magnesium hydroxide and magnesium ammonium phosphate were studied under the electron microscope. The form and texture of the barium sulphate crystals were found to depend upon the concentration of precipitant in solution. Figure 4 shows a single barium sulphate crystal grown 30 min after precipitation from 1a25 X 10e4 molar solution. The regularity

9 E

500

100

I

2

3

4

Concentration of 00

S04,

5

6

g mol/l

x104

7

Fig. 3. Conductivity of barium sulphate solutions. A experimental, B calculated from standard conductivities, C calculated from standard conductivities but corrected for concentration according to the Onsager equation.

1296

Kinetics and mechanisms of precipitations

of the crystal is typical of those grown from solution at this relatively small degree of initial supersaturation (about 12 fold). At higher concentrations, the crystals are less regular. Figure 5 shows a larger crystal about 5~ long grown from a solution in which the initial concentration of barium sulphate was 6.25 X 10e4 molar, a supersaturation of about sixty fold. The crystal form shows a pronounced rounding, while the surface outline and texture is much more rugged. In addition, there is a distinct electron penetration around the edge of the crystal because the crystal is thinner there. A smaller crystal, 3.5~ long, grown from the same solution, is shown in Fig. 6, in which there is some rounding of form but greater regularity than the example of Fig. 5. The contour fringes are thickness contours due to diffraction contrast set up because the thickness of crystal is small near the edge and increases into the centre of the crystal. Features of the diffraction contrast illustrated by Fig. 5 have been discussed in further detail elsewhere [14]. The crystals of magnesium ammonium phosphate were larger as shown in the micrograph Fig. 7. Some of the crystals were about 20~ in length. Figure 8 shows a crystal of magnesium hydroxide. This particular crystal was examined by selected area diffraction; the pattern of Fig. 9 shows clearly that it is not a single crystal, and that a large number of small crystallites have intergrown during the precipitation process. EXPERIMENTAL

RESULTS

(a) The induction period The experimental dependence of the induction time upon concentration is shown as a graph of the logarithm of the induction time against the negative of the logarithm of the mean ionic concentration. Figure 10 shows such a plot for barium sulphate; equal volumes of equivalent concentrations of aqueous barium chloride and sulphuric acid solutions were mixed and the time of appearance of the first scintillations was measured. The stoichiometric ratio of barium to sulphate ion was maintained at 1.0.

0) -3

-2

-I

0

2

3

4

[Ati secl Fig. 10. The induction period for equimolar solutions of barium sulphate and its dependence upon the average ionic concentration. 0 this investigation, X results of Dinegar and La Mer [2], 0 results of Christiansen and Nielsen. log

The figure includes the experimental results of Dinegar and La Mer[2] obtained by a similar technique. The results of this investigation and those of Dinegar and La Mer obey the equation At, = C* [{ [Ba++][S04]}-1’*]3

(7)

under the restrictions of equivalent ionic concentration, where Ati is the induction period. Christiansen and Nielsen [ 11, whose results are also shown in Fig. 1, measured the induction period from experiments in which the reagents were mixed by flow into a mixing junction. Their results are not in agreement either with the results of this investigation or with the results of Dinegar and La Mer. The latter finding is surprising because La Mer[2] has previously reported a good agreement. Christiansen and Nielsen presented their results as a graph of the logarithm of the product of the concentrations of barium and sulphate ions against the logarithm of the induction period; unfortunately La Mer misread this to be a graph showing the logarithm of the mean ionic concentration and so deduced agreement where none existed. In a second series of experiments, the ratio of the two precipitating ions was varied for a number of precipitations and the time of fhst appearance of the crystals was again measured for each experiment. The results are shown in Fig. 11 where the ratio of the concentration

1297

D. J. GUNN

3.0

0

I

2 log

and M. S. MURTHY

I 3

[At, set]

Fig. Il. The deiendence of induction period for barium sulphate upon average ionic concentration when the ratio of barium to sulphate ion concentrations varies. The ratio is shown as a parameter.

of sulphate to barium ion is shown as a parameter. It is clear that Eq. (7) is also a valid description without the restriction of equivalence of precipitating ions; the exponent is unaffected by the change in ionic ratio. These experiments were carried out at 20°C. The effect of temperature upon the time of fh-st appearance of crystals was measured by setting the temperature of the thermostatted enclosure to a predetermined value and allowing the precipitant solutions to reach a steady temperature before mixing in the enclosure. Figure 12 shows the results of these experiments expressed in a manner suggested by the form of Eq. (7). As CI is directly related to a reaction velocity constant in a way to be shown later in the paper, then an Arrhenius dependence upon temperature may be expected. Figure 12 contirms the expectation. The magnitude of the activation energy in the Arrhenius equation found from this Figure is in the region of 2.5 kCal/mol, of similar order to energy changes associated with the transfer of molecules from crystal to solution. The same experimental methods were used to investigate the induction period of magnesium hydroxide. In this case the precipitant solutions were magnesium sulphate and sodium hydroxide. Two series of experiments were carried out. In the first the concentration of precipitating ions was varied but the stoichiometric ratio of

Fig. 12. The dependence of the induction period upon temperature for two solutions of barium sulphate A 3.75 X 10e4molar. B 2.5 X lo-’ molar.

magnesium to hydroxyl ions was maintained at 1:2. The time of first appearance of crystals was measured as before. In the second series the stoichiometric ratio of magnesium to hydroxyl ion concentration and the mean ionic concentration were both varied. Figure 13 shows both sets of results. The ratio of twice the magnesium ion to hydroxyl ion concentration is shown as a parameter. The experimental results obey Eq. (8) At, = Ci [{ [Mg++][oH]‘}-“3]5

(8)

an equation of the same type as Eq. (7). The dependence of CI upon temperature for magnesium hydroxide was also investigated and the results of these experiments are shown in Fig. 14. The magnitude of the apparent activation energy is similar to that found for barium sulphate. A third sparingly soluble salt studied was magnesium ammonium phosphate. This salt was formed by adding a solution of ammonium hydrogen phosphate to a solution of magnesium sulphate and ammonium hydroxide. The induction time was measured and the results are shown in Fig. 15. Again it is evident that there is a power law dependence of the induction

1298

Kinetics and mechaaisms of precipitations 2.02

/ I.14 . -1.14

/ J-

0.925

x-o.91 - 0.09

2’s2.7

/

,

0

2 log bt,

3

secl

Fig. 13. The induction period for solutions of magnesium hydroxide. The dependence upon the mean ion& concentration is illustrated with the value of 2 [Mg++] I [OH] given as a parameter.

Fig. 15. The dependence of the induction period upon mean ionic concentration for solutions of magnesium ammonium phosphate.

(b) The

concentration of barium solutions during precipitation

in

The concentrations of barium sulphate measured in solutions during crystallization by means of the conductivity cell are shown in Fig. 16 for three solutions in which the initial concentrations of barium sulphate was rapidly reduced in the solution having the greatest initial concentration, but the solution of the least initial concentration showed a slow fall in concentration, and almost 24 hr was required to reach the saturated value.

‘G-

(c) The surface pension .6 -

3.c,

@hate

I

I

I

I

I

31

3.2

3.3

3.4

3.5

+

xldV=K-‘1

Fig. 14. The dependence of the induction period upon temperature for two solutions of magnesium hydroxide A 1 X 10e3molar. B 8.75 X 1O-4molar.

period upon the mean ionic concentration of the precipitants. The particular equation for this case is, --Ari = Ci[{[Mg+f][NH~‘][P0J}-“3]2. (9)

area of barium crystals in sus-

The intensity of light transmitted through a solution during the crystallizing period was measured by means of the photo-sedimentometer and the reading was continuously recorded on a strip chart. To relate the intensity to the true surface area of the particles the value of (Yhas to be estimated; the value of (Yis ?r2/2 for convex particles with plane surfaces and 4 for spheres. The value of ti/2 represents a lower limit for particles having concave regions in the surface structure, while the values of 4 and 7r2/2 represents the lower and upper limits of CYfor particles without concave regions in the surface

1299

D. J. GUNN

and M. S. MURTHY mE 2.0 P la

B

C ye-.-.

0

IO

20 Time,

0

5

IO

15

Time,

20

25

35

min

Fig. 16. The change with time of concentration of barium sulphate in crystallising solutions. A initial concentration 6.25 X 10e4 molar, B 3.75 X 10m4molar, C 2-5 X lo+ molar, D saturated solution.

structure. As the electron micrographs showed that the crystals are convex when small with some convex curvature to the surfaces the value of (IIwas taken to be 4. However, in the example of Fig. 5, the surface is striated and undulating even though the principal shape of the crystal is convex- the striations and undulations in the surface are not included in the estimate of the total surface area which was calculated from Eq. (1). The development of surface area with time for each of the three solutions of barium sulphate is shown in Fig. 17. (d) The particle size distributions The interpretation of measurements from the photosedimentometer requires estimates of the correction factors j3 and y. An obvious choice is to suppose that /3 and y are each equal to 1, the appropriate values for spherical particles. The approximations were examined in the light of evidence concerning the size and shape of the particles provided by the electron micrographs and information on the hydrodynamic resistance of prolate and oblate spheroids [ 151. An idealised and rounded shape of the crystals seen in the electron microscope was that of an ellipsoid;

30

min

Fig. 17. The development of crystal surface area with time for crystallising solutions of barium sulphate. A initial concentration = 6.25 X 10m4 molar, B 3.75 X 10m4molar, C 2.5 X 10m4molar.

the equations of Payne and Pelll151 give some idea of the effect of change of shape upon the drag force and therefore allow the drag force acting upon a crystal ellipsoid to be estimated. Figs. 4-6 are micrographs of barium sulphate crystals. From these and other micrographs it appeared that the crystals were rectangularelliptical in shape of length L, and breadth O-6 L when laid in the most stable position. There was evidence of a rhomboidal form to the crystal edges and it appeared that the thickness of the crystal in the centre was about 0.3 L although the photographic evidence was uncertain about this dimension. The true surface area of the barium sulphate crystal was estimated to be about 1.8 L2 and therefore the equivalent radius r of the particle was related to the length L by the equation ?r? = 1.8 L*/a where a! has been taken to be 4. This gives L - 2.5r. The motion of fluid round a sedimenting particle in the size range < 10~ is laminar with the Reynolds number of the particle much less than one, so that the long axis of the crystal might be expected to remain vertical during sedimentation because in this position the drag force is minimized. It appears reasonable to assume that the idealised crystal ellipsoid of dimensions O-6L and O-3L at the particle equator will experience a drag force similar to that experienced by a prolate spheroid of diameter

1300

Kinetics and mechanisms of precipitations

= d(O.6x 0.3L2)- 0.4L at the equator

when moving at the same velocity. From the equations of Payne and Pell it is found that the ratio of the drag force acting upon this prolate spheroid to the force acting upon a sphere of radius r moving at the same velocity is O-65, a value for p that is significantly smaller than one. An estimate of the volume of a particle requires a more comprehensive knowledge of the shape than the electron micrographs were able to provide, and for this reason it was decided to estimate the value of y from a material balance for solute in the crystallizing solutions of barium sulphate. If w is the weight of solute crystallized from unit volume of solution, the material balance gives: w - 4TPsY r-g 3 I

(r)$dr.

(10)

From Eqs. (3) and (6) this equation may be expressed in terms of y and direct experimental observations:

The weight of solute crystallized may be found from the initial and final concentrations of the solution, so that the value of y may be found by a graphical solution of Eq. (11). Equation (11) was solved for y from the experimental results obtained with two solutions of barium sulphate, one of initial concentration 6.25 X 10-4m and the second of 3.75 x lop4 m; the calculated values were y = O-65 for the first and y = 0.60 for the second. Figure 18 shows three particle size distributions obtained from solutions of barium sulphate of different initial concentrations. It can be seen that the average particle size developed during a batch crystallization in this range of concentration increases with an increase of initial concentration. Two of the size distributions show maxima, and in the case of the most concentrated solution the distribution is very narrow. Interference by convection currents with the settling of very small particles prevented the maximum of the third distribution being found.

Radius,

p

Fig. 18. Particle size distributions of barium sulphate crystals. A initial concentration = 6.25 X 10m4 molar, B 3.75 X lo-* molar, C 2.5 X 1Oe4molar.

In all three cases it is evident that the largest particles are relatively few in number, and as the largest particles are probably those that have grown for the longest period, the rate of nucleation at the start of the experiment must be relatively low. If nucleation is homogeneous at the start of the experiment, because this rate will decrease with falling concentration, nucleation in the later stages of the experiment must proceed by a different mechanism. Particle size distributions of magnesium hydroxide were also studied. Figure 19 shows the particle size distributions of two magnesium hydroxide precipitates obtained from two solutions of different initial concentration; the ratio of /3/y was set equal to 1. As for the distributions of barium sulphate the largest particles are relatively few in number, while the largest average size is developed from the solution of greatest initial strength. The narrowness of the size distribution obtained from solutions of high initial strength shown in Figs. 18 and 19 is striking. The narrow distribution is a result of the rapid diminution of

1301

D. J. GUNN

and M. 5. MURTHY

200 -

“0 ‘ii

‘:

E

0

150-

$ 100 -

50B

0

, 2

$

1 4

I

I

6

6

Fig. 19. Particle size distributions of magnesium hydroxide crystals. A initial concentration = 5 X 10mSmolar, B 1 X low3 molar.

concentration in these solutions. The rate of nucleus formation is thereby sharply reduced and therefore the crystals are of similar age and size. (e) The rates of linear growth The measurements of surface area and concentration may be used to calculate the rate of linear growth of crystals of barium sulphate, The rate of growth was calculated in the form drldt and therefore the equation relating the rate to the experimental variables is: dr --M dc dt = ypsa dt ’

(12)

The derivative dc/dt was found by graphical differentiation of the concentration-time curve. Figure 20 shows the rates of linear growth for crystals of barium sulphate as a function of supersaturation in the three solutions of barium sulphate. The rate of growth is proportional to the third power of the supersaturation for all three solutions when the supersaturation is

16’

Supersoturotion.

IO-’

CO-C,

IO-'

gmd/l

Fig. 20. The dependence of the rate of linear growth of barium sulphate crystals upon supersaturation. 0 solution of initial concentration = 6.25 X lo+ molar, 0 solution of initial concentration = 3.75 X 10e4 molar, X solution of initial concentration = 2.5 X lo-’ molar.

relatively large, but for the solution having the greatest initial concentration the rate of growth becomes proportional to supersaturation at low supersaturations. There is a less marked change of proportionality for the solution of intermediate strength, while for the solution of lowest initial concentration the dependence of growth upon the third power of concentration is maintained to low supersaturations. It is interesting to note that for the solution of greatest initial concentration the rate of growth is apparently enhanced at low supersaturations compared to the rate controlled by third order kinetics. Although the rate of growth is shown as a function of supersaturation, it is not intended to dismiss the possibility of expressing the rate of growth as a function of the concentration rather than the supersaturation of precipitant. The form of Fig. 20 is but little affected at high con-

1302

Kinetics and mechanisms of precipitations IO’

centrations because concentration and supersaturation are sensibly equal when the concentration of the saturated solution is small compared to either. However, for more dilute solutions the use of concentration as the independent variable does not give such a convenient graph on logarithmic co-ordinates so that the use of supersaturation is preferred. (f) The rates ofnucleation The rates of nucleation may be calculated from the particle size distribution and the rates of linear growth. For a crystal of size r the time at which the crystal was nucleated to give a crystal of equivalent radius that is effectively zero is given by the equation: r=

mdr JO t, dt

dt

(13)

and therefore a graph of drldt against t that starts from a late stage in growth at which dr/dt is effectively zero will give t,, the time at which crystals that grew to that particular size were nucleated. As the relationship between r and dn(r)/dr has been found, once the time of nucleation has been calculated from Eq. (13), the rate of nucleation may be calculated from Eq. (14). -=-

(14)

The integral T+Ardn(r) s T drdr

0

Fig. 2 1. The rates of nucleation of barium sulphate solutions as functions of supersaturation. 0 solution of initial concentration = 6.25 X lo-’ molar, 0 solution of initial concentration = 3.75 X lo+ molar, X solution of initial concentration = 2.5 X lo-’ molar.

initial concentration there is a marked plunge of rate within a very short time interval. In all three cases it is clear that a mechanism of nucleation that is not homogeneous rapidly grows to dominance. DISCUSSION

gives the number of crystals in unit volume within the size range of r to r + Ar, and the time interval At is the difference in the time of nucleation of a particle of size r+ Ar, and the time of nucleation of a particle of size r; the quotient of the two quantities gives the rate of nucleation at the time t + Atl2. The rates of nucleation calculated in this way are shown in Fig. 21 in a plot on logarithmic coordinates for the three solutions of barium sulphate. Each graph shows a rapid rise in the rate of nucleation but for the solution of the greatest

The significance of the induction period It has been asserted[l, 21 that the induction period can be interpreted in terms of the rate of nucleation and Christiansen and Nielsen have also taken it to be the time at which a certain frao tion of the solute has been precipitated from the solution. However, although a relationship between the induction period and the rate of nucleation is a subject for coriecture it is clear that the physical significance of the induction period is simply that some crystals have grown sufficiently large to scatter incident light.

(a)

1303

D. J. GUNN

and M. S. MURTHY

The theory of light scattering by spherical particles was given some years ago by Mie [ 161, who related a scattering coefficient K to the scattering of incident radiation by the particle. Significant scattering of incident light does not take place until the radius of the particle is comparable to the wavelength of the incident light; in fact the scattering coefficient K will reach a maximum when the radius r of the particle is equal to the wavelength of the light in the suspending medium. Because white light was employed in the experiments the average wavelength in aqueous solution was taken to be 0.5~. Scattering of light from particles of size 0.5~ should be clearly observable because the scattering coefficient is a maximum so that the induction period will be ended when the particles have grown to 0.5~ and the scattered light will appear as scintillations from the crystals. The measurements of conductivity have shown that the concentration of barium sulphate remains essentially constant during the induction period. As the rate of growth for small crystals is expected to depend only upon the concentration of precipitant and temperature, the rate of growth within the induction period should be constant. Thus if the crystal grows from zero to a size ri (just estimated to be 0.5~) during the time interval of induction Ati, the rate of growth drldt is then given by the equation:

bh_ Z = Ati'

(15)

Figure 22 is a graph showing the change of dr/dt with initial supersaturation when drldt is calculated from Eq. (15) and the experimental estimates of the induction period: the figure includes a line showing the third order dependence of drldt upon supersaturation taken from Fig. 20. The agreement of this line with the estimates of Eq. (15) is surprisingly good, providing very good evidence for the simple interpretation of the induction period as the time interval required for crystals to grow to an observable size. Growth rates in solution are often very difficult

4

c

5

Id’

16' Supersaturation,

CO-C,

g mol/l

Fig. 22. Estimation of the rate of linear growth for barium sulphate from measurements of the induction period. The full line has been taken from Fig. 20.

to determine by other means so that the simplicity of the determination of growth rates from observations of the induction period is very attractive. The growth rates estimated in this way are restricted to small crystals where the influence of diffusion is not important and the interface reaction is therefore dominant. (b) The mechanism of the interface reaction During the process of growth from solution there is a variation of electric potential over the surface of the crystal which depends upon the distribution of electric charge due to ions held in the surface regions. Ions in solution tend to diffuse towards regions of opposite charge in the surface under the influence of the electric field, however it is important to realise that the polarity of electric potential in the vicinity of a surface vacancy will be reversed when the site is occupied. The charge of an ion in the complete crystal structure is balanced by the charge distri-

1304

Kinetics and mechanisms of precipitations

bution of neighbouring ions; the charge of an ion in the surface of a crystal is not balanced, so that when a surface vacancy is filled one unbalanced state is succeeded by a state unbalanced in the reverse sense and there is a change in polarity of electric potential in the neighbourhood of the site. As a consequence a single ion migrating towards a vacant site in the crystal surface will experience a repulsive force at close proximity, particularly if the ion is small when the ionic charge is more highly concentrated and the electric field is more intense. If the ion is large, or has become large by association with other ions in solution, the field is less intense because the ionic charge is distributed over a greater volume and there is an enhancement of electric polarization within the cluster. The consequent reduction of field intensity in the vicinity of a site during growth will correspond to a reduction in the activation energy of the growth reaction. These considerations suggest that crystal growth by incorporation of ionic clusters may be kinetically favoured over crystal growth by incorporation of single ions because of the greater size of the cluster and the increased possibility of polarization of charge leading to a reduction of repulsive force when the ionic cluster and vacancy are in close proximity. In the case of barium sulphate, suppose that the process of growth takes place by preliminary ionic association in solution followed by reaction between associated ions and surface vacancies in the sequence: Ba++ + So, -+ BaSO, BaS04 + Ba++ +

[BaSO,Ba++]

(16)

the symbol c denotes the crystal phase. If the first three reactions are rapid relative to the latter two, the concentrations of barium sulphate and the associated ions will be similar to the equilibrium values. The rate of deposition of ions from solution on the crystal surface is then given by kinetic equations derived for the ionic mechanisms of Eqs. (19) and (20). Under conditions of steady state the rates of deposition N+ and N- are given by: N+ = k”[BaS04Ba++] [-]

(21)

-N- = k”[SOIBaSO,l [+I

(22)

which, on substitution for the equilibrium centrations of the associated ions, become:

con-

N+ = k’[Ba++12[S6J

[-]

(23)

N- = k’[Ba++] [S&,]“[+].

(24)

To maintain electroneutrality within the solution, the rate of deposition of sulphate ions must be equal to the rate of deposition of barium ions and therefore N+ = N- so that 9

[Ba++l [+I

[s6,1=[-1’

Thus if there is a preponderance of barium ions in solution the crystal surface will include a preponderance of vacancies for sulphate ions. The form of the dependence of the concentration of cation and anion surface vacancies upon ionic concentrations may be inferred from a study of conditions in the solution at saturation:

(17) BaSOl(c) s

BaSO, + S
(25)

[Ba++l+ [SoJ + [+I + M.

(26)

(18)

[BaSO,Ba++] + [-] + BaSO,(c)

(19)

-[S04BaS041 + [+I + BaSO,(c).

(20)

The symbols [-] and [+I denote concentrations of surface vacancies for cations and anions and

As the concentrations of barium and sulphate ions in the solution are related: [Ba”]

[S&j

= solubility product.

Then on writing down the equilibrium condition for Eq. (26):

1305

D. J. GUNN

Da++1[SO41[+I C-l = [BaSWc)1

k

and M. S. MURTHY

(27)

it is clear that [+I[--] = constant. This equation then gives

in conjunction

(28) with Eq. (25)

(29)

(30) where c is a constant of proportionality. The rates of ionic deposition N+ and N- which are equal to N, the molecular rate of deposition of barium sulphate are then given by N+ = N- =

k[d(

[Ba”]

[S0,])13

= N.

(31)

The rate of molecular deposition N is directly proportional to the growth rate drldt, so that this equation agrees with the form of the dependence of drldt upon the mean ionic concentration found from experiment. Furthermore if Eq. (3 1) and Eq. (15) are combined, then: Ati = riki-‘[{ [Ba++] [SGJ}-““I”

(32)

an equation which becomes Eq. (7) on setting riki-’ ---*C,; therefore the theoretical equation for the rate of deposition of barium sulphate agrees both with experimental measurements taken during crystallization, and growth rates deduced from the induction period. Similar mechanisms for the crystallization of magnesium hydroxide, and magnesium ammonium phosphate which lead to the rate equations associated with Eqs. (8) and (9) may also be derived. The high degree of association for the growth reaction of magnesium hydroxide suggested by Eq. (8) is probably due to the small size of the magnesium and hydroxyl ions. The ions in the phosphate complex are larger so that a relatively low degree of association will lead to

favourable kinetic conditions for growth, and this is reflected in Eq. (9). (c) The interaction between diffusion and growth In the early stages of growth when crystals are very small the concentration of precipitant everywhere in an agitated solution is essentially constant. As the crystals grow in size significant concentration gradients are set up in the neighbourhood of each crystal. Some of the features of the interaction between diffusion and growth may be illustrated by supposing that a steady diffusion field having spherical symmetry surrounds each crystal. The field equation is (33) The crystals are not spherically symmetric, but it is supposed that near the crystal surface the accommodation between diffusion and growth is met by the condition:

q=-Ddc dR

=

ka,(ci - c,)~ at R = r

(34)

(the equivalent radius of the crystal). This equation is one boundary condition for Eq. (33); the second is given by the condition that because the suspension is dilute, the concentration of precipitant far from the crystal surface is equal to the bulk concentration co. With these two conditions the solution to Eq. (33) is: (35) Near the surface becomes:

of the crystal this equation

(36) When the radius r of the crystal is small the right hand factor is diminished and therefore the concentration of precipitant at the interface, ct, is similar to the bulk concentration co, while when

1306

Kinetics and mechanisms of precipitations

r is large ci is quite different from co so that ci approaches c,. When the crystal is relatively large concentration gradients in its vicinity are significant. Although Eq. (35) may give an acceptably accurate picture of steady diffusion and growth, the growth may not be uniform and steady. A growth perturbation on the surface of the crystal as shown in Fig. 23 may not be absorbed into a planar growth. Consider a growth perturbation of magnitude dB caused by some local variation in, say, precipitant conditions. The rate of growth q of the tip of the perturbation protruding into the diffusion field is initially enhanced at a rate dq/dR obtained from Eqs. (34) and (35) in the following way:

(37) -dq

=

zg!z

(Ci

-

45.

cm

Because of the increase of concentration of precipitant with distance from the crystal surface, the perturbation will show an initial growth surge. If the solubility of the precipitated material is large the enhanced rate of growth can be supported by a relatively small diffusion field

Fig. 23. Growth perturbation on the surface of a crystal.

around the perturbation and therefore crystal growth. of a dendrite pattern will develop. On the other hand, if the solubility of the precipitated material is small, the enhanced rate of growth will set up extensive transients in the diffusion field so depleting the supply of precipitant in the vicinity of the perturbation that the initial growth surge dies away, to be followed by a growth surge at a point on the crystal surface where conditions for growth have become more favourable. The crystal of Fig. 5 shows the effect of the surges in the minor growth regions that terrace the surface of the crystal. The pronounced rounding of the contours of the crystal is due to the surrounding diffusion field showing that the pattern of deposition of crystal material is strongly influenced by the diffusion flux towards the surface of the crystal. Conditions for dendrite growth are more favourable at higher supersaturation. Figure 24 is an electron micrograph of a barium sulphate crystal grown from a solution in which the initial concentration of barium sulphate was 0.025~2. The dendrite pattern is more significant because an increased rate of growth of a dendrite spur can be supported by a smaller diffusion field due to the higher concentration. The dendrite growths are better defined and more regular than those found at lower concentrations. An enlarged view of the crystal surface formed at lower supersaturations is shown in the electron micrograph of Fig. 25 that reveals crystal growth developing in nodular regions of about 500 Angstroms in extent. The marked roughness of the surface caused by these minor dendrite excursions is a significant increase in the area of crystal surface exposed to the solution, but the increase, as discussed earlier, is not registered by the photo-extinction method. This low estimate of the surface area will cause an increase in the apparent rate of growth calculated for Fig. 20. The apparent enhancement of the rate of reaction under diffusion-controlled conditions is probably due to the low estimate of true surface area. A similar point concerning the effect of surface

1307

D. J. GUNN

and M. S. MURTHY

roughness has been made by Bennett and Fentiman[ 171 who found that the rate of growth of sucrose crystals having roughened surfaces was proportional to the surface area determined by krypton adsorption, and not to the geometric surface area of the regular crystals. When the crystal is small the surrounding diffusion field formed during growth is limited in extent and the concentration of precipitant near the surface of the crystal is substantially equal to the bulk concentration. The rate of interfacial reaction is not affected by diffusion under these conditions; therefore the growth rate is described by Eq. (32) and appears on Fig. 20 as the straight line of slope 3 in the high concentration region. As the crystal grows in size, the diffusion field extends and intensifies leading to growth surges and a rate of linear growth that is higher than would be expected for orderly crystal growth under reaction control. The effect appears to be more important for the larger crystals. These separate modes of growth may be combined into a single rate equation. The crystals grown from the solution of initial concentration 6.25 X low4 molar have a very narrow size distribution about the average value of 2.5~. The dependence of the linear rate of growth upon concentration for crystals grown from this solution shown in Fig. 20 may be expressed by the equation: j+ = 0~0002 (co - c,) + 30000 (co -c&3

(38)

which is shown as the upper curve on Fig. 20. As crystals growing in the diffusion controlled region do not change a great deal in size because of the relatively low supersaturation, it can be inferred that a general form of Eq. (38) should be: dr z = k,(c, - c,)r + /&(co - c&3.

(39)

When this equation is related to Eq. (38) which expresses the growth equation for crystals which have a terminal average size of 2*5~.c,Eq. (40) is obtained:

$ = O-8@,,- c,) + 3OOOO(c,, - ~3~.

(40)

Equation (40) adequately represents Eq. (38) because the value of r does not differ much from 2.5~ in the diffusion controlled region. Further, the shape of the growth curves on Fig. 20 for solutions of lower initial supersaturation is explained by Eq. (40) because the growing crystals are smaller in size, and of wider size distribution than crystals grown from the solution of highest initial concentration. Thus Eq. (40) describes the phenomena of diffusion enhanced growth and reaction-controlled growth, and as far as can be ascertained, gives a good description of the experimental results for the barium sulphate solutions. (d) Secondary nucleation The particle size distributions of the barium sulphate and magnesium hydroxide precipitates show maxima in the middle of the ranges of the distribution. If nucleation of crystals occurs only in the homogeneous solution phase throughout the crystallization, the rate is bound to fall from the time of mixing because the rate of nucleation is a monotonic increasing function of concentration, and the concentration of precipitant in solution decreases as the crystallization proceeds. Because the rate of nucleation does not decrease, but rather increases with time from the start it is clear that another mechanism of nucleation becomes dominant. Under some conditions existing crystals may induce the formation of new crystals by shedding crystal dust from the surface, or by crystal fragmentation following a collision as discussed by Mason and Strickland-Constable [ 181. A general term for this phenomenon is secondary nucleation. Many of the crystals observed in this investigation may have been formed by a secondary mechanism. The probability of a small crystallite being shed from the parent phase may depend upon the supersaturation at which the parent phase is formed. If the supersaturation is small, the low velocities of crystal growth may allow a higher

1308

Kinetics and mechanisms of precipitations

degree of crystalline perfection than is possible for a crystal formed at a higher supersaturation. A higher density of dislocations may be expected for crystals formed at higher supersaturation leading to a greater probability of damage by internal stresses. The survival of a small crystallite shed from the parent phase will depend upon the conditions for growth in the solution. If the supersaturation is small the crystallite will dissolve because of its small size and enhanced solubility, but if the supersaturation is large the crystallite will develop and grow. The effective rate of secondary nucleation may depend upon supersaturation because of these two considerations; an increase of supersaturation should increase the rate of secondary nucleation. The rate of shedding of crystallites may depend upon the total surface area of crystals present; if so, the relationship is probably a linear one and the equation giving the dependence of the rate of secondary nucleation upon conditions in the solution may have the form: $

= a (function (supersaturation)).

(41)

The term (l/a)(dn,/dt) will depend upon supersaturation alone, and therefore should be a monotonic increasing function of supersaturation. Except in the very early stages of crystallization it appears that the rate of nucleation from the more concentrated solutions of barium sulphate is dominated by secondary nucleation and therefore (dnldt) - (dn,/dt). The variation with time of (lla)(dn/dt), and hence of (l/a)(dn,/dt) by this approximation, has been calculated as (l/u)(An/At) for the barium sulphate solution of initial concentration 6.25 X 10m4molar. The graph is shown as Fig. 26. The value of (l/u) (dn,/dt) increases from the start of the experiment although the concentration of precipitant falls, so that the rate of secondary nucleation is not apparently proportional to the surface area of the crystals in suspension. A second possibility was found to arise from

50

“0 x

0

‘”

91

Y

30-

E 0

Sk 20-

0

-lo IO-

0

II

I

I

I

I

I

IO

Time,

II

I

I 20

set

Fig. 26. The change of the rate of nucleation for unit area of crystal surfaces with time when the initial concentration of barium sulphate is 6.25 x 10m4molar.

observations of crystals under the electron During observations under the microscope. electron microscope, the intensities of the electron beam were minimized so as to eliminate the possibility of crystal damage by electron bombardment, and therefore damage observed in a number of instances must have occurred during the period of crystal growth. Figure 27 shows a barium sulphate crystal which has been damaged, and the damage followed by a period of growth upon the fractured surface. Figure 28 shows a crystal split in two; the cleaved portions remain close together suggesting that the damage has been sustained near the time of sampling. Figure 29 may show unusual damage to the crystal, or it may show an odd example of inhibited growth. Is it possible that mechanical agitation of the solution caused crystal damage? The effect of mechanical agitation upon a crystal 5~ long suspended in solution should be small because the fluid flows in a motion that is laminar relative to the crystal and the inertia forces are very small so that the possibility of damage sustained in collision is likely to be small. However, there is evidence of dislocations in barium sulphate crystals [ 151. When the rate of linear growth is high there will be a high density of dislocations

1309 CES Vol. 27 No. 6-H

40-

D. J. GUNN

and M. S. MURTHY

in the crystal structure, and subsequent movement of dislocations may cause concentrations of stress that are sufficient to fracture the crystal as in Figs. 27 and 28. One further possibility is raised by Fig. 30. This shows a growth whisker emerging from the shoulder of a barium sulphate crystal. Growth and subsequent fracture of whiskers is a possible source of crystal nuclei [ 181. The incidence of crystal damage is not likely to be proportional to the surface area of the crystals. In dilute suspensions under laminar conditions the possibility of collision between crystals is small, and therefore the probability of damage to a single crystal should be independent of the presence of other crystals. Hence the total number of instances of damage leading to fracture should be proportional to the number of crystals. This would be the case if the principal mode of damage to crystals was caused by internal stresses. The form of relationship might be:

dn,_- n (function (supersaturation)). dt

d

(42)

If this equation holds, (1 /n)(dn,/dt) should be a monotonic increasing function of supersaturation. The value of (l/n)(dn/dt) has been calculated for three solutions of barium sulphate under conditions of domination by secondary nucleation so that (l/n)(dn/dt) - (l/n)(dn,/dt). The dependence of (l/n)(dn,/dt) upon supersaturation is shown in Fig. 3 1. Clearly (l/n) (dn,/dt) increases monotonically so that Eq. (41) fulfils this important condition. The figure also shows that the degree of supersaturation has an important influence. On the logarithmic coordinates of Fig. 31 the slope of the graph is 2.5, and therefore the full form of Eq. (41) is: dn -$ = k#(C, - C,)2’“.

16’

(43)

The crystals of Figs. 27-30 were examined after remaining in contact with solution for 30 min. At an earlier stage of growth the shape of

IO-’ IO-

IO-’

Supersaturation,

Co-C, gmd/k

Fig. 3 1. The dependence of (l/n)(dn,/dt) upon supersaturation for three solutions of barium sulphate. 0 initial concentration of solution = 6.25 X 10m4molar, 0 initial concentration of solution = 3.75 X 10m4molar, X initial concentration of solution = 2.5 XFOW4 molar, 0 calculated when both primary and secondary nucleations are considered from Fig. 33.

the crystals was found to be significantly different. If the crystals are taken from solution at a time before the visible onset of growth and examined under the electron microscope the crystals appear as irregular fragments that evidently grow into regular crystals in the later stages of crystallization. Figure 32 shows crystals taken from a barium sulphate solution of initial concentration 2.5 X 10e4 molar 25 set

1310

Kinetics and mechanisms of precipitations

after mixing. The induction period for this solution is 75 set so that the crystals in Fig. 32 appear at a stage considerably before they could be detected by scintillations in the solution. From Fig. 21 it may be seen that the rate of nucleation is increasing at this stage of crystal formation, and for these irregular crystals it is possible that internal stresses associated with the presence of dislocations formed during growth will cause a rupture of the crystal so providing nuclei for further growth. Figure 31 shows that for a solution of concentration 2.5 x 10m4molar the frequency of secondary nucleation is once for each crystal in every 100 set, while at a concentration of 6.25 X 10m4 molar the frequency has increased to once for each crystal every two sec. (e) Primary and secondary nucleation Although the rate of nucleation is apparently dominated by the secondary mode in the later stages, obviously crystals must first be formed by the primary mode before secondary nucleation can take place. Hence in the very early stages of crystallization the rate of primary nucleation must be important, and indeed must dominate at first, The total rate of nucleation dnldt is the sum of the rate due to the primary mode, and the rate due to the secondary mode given by Eq. (43). In the early stages of crystallization, when the concentration of precipitant is constant, dnldt will be given by the equation:

$=

condition a graph of ln n against t will be linear of slope C, and intercept In (CJC,). Figure 33 is a graph of In n against t for the three barium sulphate solutions. For a considerable region of each graph there is a clear proportionality between In n and t. For large n there is pronounced departure from linearity because the concentration of precipitant falls, and therefore Eq. (45) no longer applies while for small n, (C&J n is not large compared to 1 so that although Eq. (45) applies a linearity of In n with t is not found in this region. This last deviation can be corrected by rewriting Eq. (45): ln[(l+~n)~]=C,tfln~.

(46)

By taking the logarithm of ((C,/C,)+n) when Cl/CZ is calculated from the intercept of Fig. (33), Eq. (45) may be tested to low values of n.

IO6

T

E 0

IO5 ;

C,+C,n

10.

with the initial condition n = 0 at t = 0, where cl and cz are constants which represent the primary and secondary modes of nucleation. The solution to this equation is: ln(1+2n)

= C,t

40)

(45)

I

0

a relationship that is valid from the start of the experiment for the time in which the concentration of precipitant is sensibly constant. As crystals grow in number the value of C.&C, becomes large compared to 1 and for this

I

0

I IO

I

I

20

30

I 40

I

I

I

50

60

TO

I 200

I loo

Time,

I(b) I

300(c)

set

Fig. 33. Primary and secondary nucleation in solution; Eq. (43) as a description of nucleation in barium sulphate solutions.

1311

D. J. GUNN

and M. S. MURTHY

The important corrections are shown as displacement arrows on Fig. 33. Equation (45) is seen to be a good description of the experimental results presented in this way, although there is some deviation of the results for the solution of initial concentration 6.25 X 10e4 molar. The value of C,/C, is the same for each solution because there is a common intercept. This shows that the concentration dependence of the primary mode of nucleation is the same as for the secondary mode, already shown to be proportional to (cO- c,)~.~ so that the complete form of Eq. (44) for solutions of barium sulphate is: $

= kl(CO- cJ2.5 + k,n(c, -c&2.5.

D

f g H

I IO

k, k’, k”, ki kl, k2 kD, kR

(47)

It may be shown that his analysis of primary and secondary nucleation is consistent with the analysis of secondary nucleation presented in the last section. The value of cz for each solution may be obtained from the slopes of Fig. 33. The constants for secondary nucleation calculated from the slopes which are equal to (l/n)(dn,/dr) are shown on Fig. 31, and are clearly consistent with the information presented in that figure.

L

A4 N,N+, Nn n(r) n,

Acknowledgement-The authors wish to acknowledge financial support for this study from UKAEA, Aldermaston. 4 r

NOTATION

surface area of particles in unit volume of suspension projected over a plane normal to the incident light a surface area of particles in unit volume of suspension a, ratio of surface area of the crystal to the surface area of a sphere of radius r Cl, C2, Ci constants c concentration of precipitant ci concentration of precipitant at the crystal-solution interface co concentration of precipitant in the bulk of solution c, concentration of precipitant at saturation A,

ri R t

4l V W

diffusion coefficient drag force acting upon particle acceleration due to gravity height of fall from surface of liquid to photosedimentometer light beam intensity of light passing through photosedimentometer cell intensity of light passing through photosedimentometer cell when filled with clear solution reaction velocity constants reaction velocity constants for primary and secondary nucleation reaction velocity constants for diffusion controlled and reaction controlled growth length of light path in photosedimentometer cell molecular weight rate of deposition of precipitant, cations and anions number of particles in unit volume number of particles in unit volume of radius greater than r number of particles in unit volume formed by a secondary mechanism diffusion flux characteristic dimension of particle. The area of the particle projected onto a plane normal to incident light is nr2 size of particle at close of induction period radial co-ordinate time time of nucleation velocity weight of solute crystallised from unit volume of solution

Greek symbols

1312

a p

defined by condition: true surface area of particle = cvirr2 ratio of the drag force acting upon a particle of size r to the drag force

Kinetics and mechanisms of precipitations

y

acting upon a sphere of radius r moving at the same velocity ratio of volume of particle of size r to the volume of a sphere of radius r

At, A p ps p

induction period difference quantity density of solution density of solid viscosity of solution

REFERENCES CHRISTIANSEN J. A. and NIELSEN A. E.,Acta. Chem. Scand. 19515 673. LA MER V. K., Ind. Engng Chem. 1952 44 1270. TURNBULL D.,Acta Merallurgica 1953 1684. WALTON A. G. and HLABSE T., Talanta 1963 10 601. [5] NIELSEN A. E., Kinetics ofPrecipitation. Pergamon 1964. [6] DOREMUS R. H.,J.phys. Chem. 1958 62 1068. [7] O’ROURKE J. D. and JOHNSON R. A.,Anal. Chem. 1955 27 1699. [8] NANCOLLAS G. H. and PURDIE N., Trans. Faraday Sot. 1963 59 735. [9] COLLINS F. C. and LEINEWEBER J. P.,J. phys. Chem. 1956 60 389. [lo] NANCOLLAS G. H. and PURDIEN., Quart. Rev. 1964 18 1. [ 1 l] KAYE B. H., Symposium on Interaction between Fluids and Particles. Institution of Chemical Engineers, London 1962. [12] BRANSOM S. H., DUNNING W. J. and MILLARD B., Disc. Faraday Sot. 1949 5 83. 1131 GLASSTONE S., Textbook ofPhysical Chemistry. Van Nostrand 1948. 1141 GUNN D. J. and MURTHY M. S., (to be uublished). jl5j PAYNE L. and PELL W. H.,J. Fluid. MeEh. 1960 i 529. [16] MIE G.,Ann. Phys. 1908 25 377. [17] BENNETT M. C. and FENTIMAN Y. L., Symposium on Industrial Crystallisation, Institution of Chemical Engineers, [l] [2] [3] [4]

London 1969. [18] MASON R. E. A. and STRICKLAND-CONSTABLE

R. F., Trans. Faraday Sot. 1966 62 455.

Resume-Les auteurs Ctudient la cinetique de la croissance des cristaux de sels peu solubles. Une etude detaillee est faite de la cristallisation du sulphate de baryum; on mesure la concentration de la solution cristallisante et la surface du precipite en tours de cristallisation ainsi que la distribution de la taille des particules a la fin de la periode de croissance. Les auteurs mesurent la periode d’induction pour 3 sels peu solubles a differentes concentrations; on remarque que le taux de croissance du sulphate de baryum presente un dependance de 3ieme ordre a la concentration quand les cristaux sont de petite taille. Pour des cristaux plus grands et a de basses sursaturations, la dependance est de lier ordre. Les auteurs ont utilist les mesures de la concentration de la surface et de la distribution de la taille des particules pour calculer les taux de nucleation primaire et secondaire du sulphate de baryum. Les auteurs presentent d’autres donndes de la nucleation secondaire et montrent qu’elles correspondent aux mesures cinttiques. Le developpement de la forme du cristal est lie a la diffusion dans la solution ainsi qu’a la reaction de surface. I1 est demontre que la petiode d’induction a un rapport simple avec le taux de croissance des cristaux de petite taille. Zusammenfassung- Die Kinetik des Kristallwachstums einer Anzahl schwerloslicher Salze wurde untersucht. Die Kristallisation von Bariumsulfat bildete den Gegenstand einer mehr eingehenden Untersuchung; die Konzentration der kristallisierenden Losung und die Obetlllche des Niederschlags wurden bei fortschreitender Kristallisation gemessen, und die Teilchengrossenverteilung wurde am Schluss der Wachstumsperiode gemessen. _ Die Induktionsperiode wurde fur drei schwerliisliche Salze bei einer Anzahl verschiedener Konzentrationen gemessen. Es wurde festgestellt, dass die Wachstumsgeschwindigkeit von Bariumsulfat eine Abhanaiakeit dritter Ordnung von der Konzentration aufweist wenn die Kristalle klein sind. For gt%ssereKristalle bei niedrig& Ubersattigungen erwies sich die Abhangigkeit als erster Ordnung. Die Messungen der Konzentration, Oberllache und Teilchengriissenverteilung wurden verwendet zur Errechnung der Geschwindigkeiten primarer und sekundarer Kernbildung von,,Bariumsulfat. Es wird weiteres Beweismaterial in bezug auf sekundiire Kembildung dargelegt und Ubereinstimmung mit den kinetischen Messungen festgesiellt. Die Entwickluna der Kristallform steht in Beziehung zur Diffusion in der Losung und der Oberflachenreaktion. Es bird gezeigt, dass die Induktionsperiode einfach zu der Wachstumsgeschwindigkeit kleiner Kristalle in Beziehung stehen konnte.

1313