lt. H. D o n ~ a i u s
10G8
Vul. 62
PRECIPITATION KINETICS OF IONIC SALTS FROM SOLUTION BY R. H. DOREMUS General Electric Research Laboratory, Schenectady, N . Y . Received A p r i l 4, 1968 Mechanisms for ionic crystal growth from solution are discussed and compared t o experimental measurements of salt precipitation rates. In the systems considered the crystal growth rate is shown t o be controlled by an interface process, rather than by bulk diffusion of solute. An adsorbed surface layer on the growing salt particles is proposed as the first stage in crystal growth from solution, and this proposal is shown to be consistent with the experimental results. It is concluded that in the systems considered nucleation occurs rapidly and the number of salt particles is constant during the measured growth period, and that factors as well as solute supersaturation influence the nucleation process.
Introduction The growth of salt crystals from solution is a fascinating process. To learn something of its nature one can use a variety of techniques. The rate of crystal growth can be measured by following the disappearance of salt from a supersaturated solution, and some information about the crystal growth process can be deduced from such kinetic studies. I n this paper mechanisms for the precipitation of ionic salts from solution will be discussed and compared t o certain existing experimental measurements on crystal-growth rates. The crystallization of salts from solution will be treated as a nucleation and growth process. Nucleation of discrete particles throughout the solution occurs when the solute supersaturation exceeds a critical value, and these nuclei then grow by transport of ions through the solution t o the particles. I n this paper isothermal growth of relatively insoluble salts from dilute aqueous solutions will be considered, and nucleation on the vessel walls will be neglected. It will be assumed that the particles are spherical and that the nuclei form in a very short initial period and consume a negligible amount of solute. After this short period the number of precipitate particles remains constant, so the depletion of solute that is measured experimentally occurs entirely by growth of the particles. These assumptions will be justified by subsequent consideration of experimental data. DifYusion or Interface Control.-The growth of salt particles may be controlled by diffusion of ions through the solution, by a process a t the particlesolution interface, or both of these mechanisms may be important. Turnbull' and Humphreys-Owen2 derived the growth equation for a single particle growing in a solution of infinite extent when both diffusion and an interface reaction are important, and Frisch and Collins treated this case when a number of particles compete for s01ute.~ For the present discussion the result of the latter authors for particles of negligible initial radius can be expressed in the form
p is the salt density in concentration units, t is the time after nucleation, D is the solute diffusion coefficient, b is an interface growth coefficient, W is the fraction of total precipitation and Rf is the particle radius when precipitation is complete (TV = 1). In terms of the average radius R of N particles per unit volume, JV is given by
TI' =
4sNR3p 3(Cm - CO)
In the derivation of eq. 1 the solute flux J per unit area of particle surface associated with the interface process was assumed to be equal t o b(C, - Co), where C, is the molar solute concentration in the solution a t the particle surface. In general J will be a more complicated function of (Cr - Co), but for this discussion of diffusion or interface control the simple form will suffice. From eq. 1 the growth is controlled by an interface reaction when D/Rr>>b; in this case C, = (C, - Co)(l - W ) Co, which is the average solute concentration throughout the solution. If D/Rf>b, and one can conclude that the rate of crystallization for these results is entirely controlled by an interface reaction. Furthermore the solution is often stirred during crystal growth experiments, so in this case the transport of ions in the solution is even more rapid than for an unstirred solution. Thus for these conditions the solute concentration in the solution at the particle surface is always (C, - Co)(l - W) Ca, and in the following discussion we need t o treat only the interface reaction t o describe fully the crystallization kinetics. Large individual crystals growing from concentrated solution have been studied under the microscope; the solute concentration around the crystal was determined from interference fringes due to In this expression C, and Coare the initial and equi- refractive index diff erences.7-9 In these experilibrium molar solute concentrations, respectively, (4) C. Zener, J . A p p l . P h v s . , 20, 950 (1949). '
(1) D. Turnbull, Acta M e t . , 1, 764 (1953). (2) S. P. F. Huniphreys-O\\~en,Proc. Roy. Soc. ( L o ? ~ d o n )6197, , 218
(1949). (3)
H.L. Frisch and F. C. Collins, J. Chem. Phys., 21, 2158 (1953).
+
+
(5) C. Wert and C. Zener, %bad.,21, 5 (1950). (6) H.Markowits, qbid., 21, 1198 (1950). (7) W. F. Berg, Proc. Roy. Soe. (London),6164, 79 (1938). (8) C. W. Bunn, Dzsc. Faraday Soc., I , 132 (1949).
Sept., 1958
PRECIPITATIOS KINETICS OF IONIC SALTS F~OM SOLUTION
ments the crystals were so large that the term 6R was appreciable. Thus solute diffusion in the solution was important in the crystal-growth process, and the solute concentration a t the particle surface was less than that a t a distance from the particle. Indeed the two latter results mere found in these studies and the authors correctly concluded that both diffusion in the solution and an interface reaction were contributing t o the crystal-growth kinetics. In this present study these experiments will not be considered further, and attention will be focused on the growth of smaller crystals for which the interface process is controlling. Equation 1 has been compared t o experimental results on the growth of barium sulfate from solution by Collins and Leineweber’o; their factor y is equal t o D / b in the present notation. In comparing the equation t o their results, Collins and Leineweber assumed that 1 6 = ij orvi
(3)
where a is an adsorption probability, v is the diffusional displacement frequency and ,s is the mean diffusional displacement. Then
This result implies that D enters into eq. 1 even for an interface-controlled process, since s is a constant. Thus the difference between a diffusion or interfacecontrolled process will appear only in the value of a(a = 1 and a < l , respectively). It will be shown below that the interface process is quite complex, so that eq. 3 describes b artificially, with all its functional variation coming in a. There is no reason t,o associate the frequency and displacement involved in bulk diffusion with the interface process; therefore it is misleading t o describe b by eq. 3, and the form of eq. 1is preferable to show the separation of the diffusion and interface processes.
Derivation of Growth Equations The functional dependence between J and C, Co will now be derived from specific models for the interface-controlled growth process. These models are preeented as plausible rat8ionalizations of the kinetics of precipitation found experimentally, and certain aspects of them will doubtless need modification as more experimental data become available. We will assume that the first step in conveying an ion from solution into a growing crystal lattice is adsorpt,ion onto the crystal surface. An adsorbed surface layer has often been invoked to explain crystal-growth results and many experiments have shown t’hat,such a layer is Wheii an ioii is xdsorbed into t.his layer from aqueous solution it may be partially or completely dehydrated. 19) G . C . Kreuger and C . W. Miller, $7. Cheni. f ’ h y s . , 21, Y O 1 8 (IY53). (10) F. C. Collins and J. P. Leineweber. THISJOURNAL,60, G80 (1950). (11) R. Marc. %. p h y s i k . Chem.. 1 9 , 71 (1912). ( 1 2 ) R. BI. Barrer, “Diffusion in and through Solids,” Caiiibridge L‘niv. Press 1941. pp. 337-381. (13) C. W. DavieR and A. L. Jones, T ~ a n s Faraday . Soc., 51, 812 (55). (1!)14) C , W. Swrii, Acta A r e ( . , S, 301 (1955).
1069
For a one-one electrolyte solution containing only the precipitating ions the concentrations of anions and cations in the layer should be virtually equal, although the concentrat ions can differ very slightly due t o preferential adsorption. We will assume that the number of ions available for reaction in the adsorbed layer is proportional to the number striking the surface less the number leaving it. Then the effective solute concentration in the surface layer is ka(Cr - Co) for both ions, where IC, is a surface adsorption coefficient which is independent of ionic concentration. If the electrolyte is twoone, such as A2B, the surface layer concentration of A is 2ka(C, - Co) and that of B is k a ( C r - CO). There is general agreement that crystal growth proceeds by incorporation of molecules at kinks in a growth step on the crystal surface.15 In the growth of ionic salts from solution the ions of opposite charge must combine stoichiometrically a t some stage in the process to maintain electrical neutrality in the crystal lattice. Two ways this ion grouping can occur are: (A) on the particle surface to form salt “molecules” which then diffuse to growth steps; (B) by alternate incorporation of oppositely charged ions directly from the adsorbed layer at a kink in a growth step. Therefore two possible models for t8heinterface process consist of the following stages. Model A : (1) Adsorption of individual ions from solution into n surface layer; ( 2 ) combination of these ions into a single uncharged salt “molecule” on the crystal surface; (3) incorporation of this molecule into the crystal lattice. Model B : (1) adsorption into a surface layer as for A; ( 2 ) alternate incorporation of oppositely charged ions into the crystal lattice. A discussion and derivation of growth equations for these two models follows. At first a solution with only the precipitating ions present, such as is formed by mixing equivalent amounts of barium hydroxide and sulfuric acid to produce a barium sulfate precipitate, will be considered, and then the case of noli-stoichiometric ion concentrations will be treated. Model A.-In this model the adsorbed ions combine in the surface layer to form neutral salt “molecules” that can fit into the crystal lattice. This step may involve further dehydration of the ions if they have retained any water of hydration in the surface layer. A variety of kinetic laws might apply for this model, depending upon the relative rates of the reaction to form molecules, the decomposition of the molecules back to ions, and the incorporation of the moleculee into the crystal. I n certain of the experiments to be discussed the flux J was found to be proportional to (C, - C0)3 for one-one electrolytes and (C, - C0)4for two-one electrolytes. To explain these results it is necessary to postulate a rather unusual process. Thc only justification put forward for this postulntioii is that it is consistent with the experimental data. When ions combine in the gas phase a third particle takes part in the reaction to stabilize the resulting molecule.18 We will assume that an (15) W. K. Burton, N. Cabrera and F. C. Frank, Phil. Trans. Roll. sol,., a43, 299 (1951). (16) E. 4. Moel~vyn-Hrrpli~s “Kinetics of Reaction3 in Solution,“ 2nd ril , Ovfoid Pic.\?. New York, N. Y., 1047, p. 1%
R. H. DOREMUS
1070
additional adsorbed ion also takes part in the surface combination process to form a salt molecule. Possibly this additional ion stabilizes the molecule by removing any excecs water of hydration from the combining ions. Then the rate of formation of the surface molecules will be proportional to Ea3(Cr - C0)3for a one-one electrolyte and ka4 (C, - C,J4for a two-one electrolyte. Furthermore we will assume that the rate of incorporation of surface molecules a t the growth steps is a relatively rapid process, so the over-all flux of ions into the crystal is controlled by the rate a t which surface molecules are formed. Then for one-one electrolytes
'"
-N A k,kSs( Cr - C O ) ~
-=
dt
(5)
in which N is the number of molecules per unit volume with average surface area A , and k , is the rate coefficient for the combination process. Thus
For two-one electrolytes in a
where b = k,ka8. similar way
J = b(Cr
- C0)4
(7)
where b = 4k,ka4; in both these equations b is independent of ionic concentration. Model B.--In this model the adsorbed ions collide directly with a kink in a growth step and are separately incorporated into the crystal lattice. Thus the rate of consumption of solute is proportional to the product of surface concentrations of the ions, or dCr
dt
-NAkakB2(Cr- C O ) ~
=
for a one-one electrolyte and
dC, = dt
-4NAksk,a(Cr - Co)a
for a two-one electrolyte, where k, is a rate coefficient which is constant with ionic concentration. This coefficient includes such factors as the frequency of collision of adsorbed ions with the kink and the probability of incorporation as a result of such a collision. When the equations for Model B are combined the results are b(Cr - Co)'
J
(8)
for one-one electrolytes and J
=
b(C, - C0)3
(9)
for two-one electrolytes, with b equal to k,ka2 and 4kska3,respectively. For both models the differential equation of growth is J_ = dR _ p
dt
- b(Cr -
- Co)" = P
C O ) ~-( WIm ~ P
(10)
where b and (C,,l - C o ) m are time-independent and m = 1, 2, 3 or 4 depending upon which equation for J applies. The integrated result form = l is eq. 1 above with Rf/D = 0, and for m = 2 a closed solution is possible. When m = 3 or 4 analytical integration of eq. 10 is difficult, but, the result can be evaluated numerically. From eq. 10 one can see that in the initial stages of growth when W is small dR,ldt is constant and TV is proportional to t 3
Vol. 62
for all values of m. An equation similar t,o eq. 10 with m equal to three or four has previously been compared t o data on ionic crystal growth. l7.18 In the above treatment the effective growth area was implicitly assumed to be proportional to the surface area of the growing particles. It is possible that after some stage of growth the effective growth area remains constant, rather than increasing with particle surface area. Then the particle surface area A must be replaced by an equivalent surface area A' which is constant with time. The growth equation can be expressed as
which integrates to 1
(1
- W)rn-l
1 - (1 - W y-1 ( m - l)NA'b(Cm - Co)tn-'(t - t ' )
(12)
where t' and W' are the time and fiaction of reaction, respectively, when the effective growth area becomes constant. Often in studies of ionic precipitation from solution seed crystals are added to a slightly supersaturated solution. In this case crystal growth takes place on the seed crystals and nucleation of salt particles is unnecessary. If the seed crystals are large enough their surface area will remain nearly constant throughout the deposition process, and eq. 11 will be valid with NA' the total surface area of the seed crystals per unit volume of solution. The integrated result is then 1
(1
- wp-1
= ( n ~-
l)NA'b(Cm
- Co)"-'t
(13)
Crystal growth experiments frequently involve mixing two soluble salts, such as barium nitrate and sodium sulfate, to give a precipitate. In this instance the above analysis is correct only if the initial concentrations of the precipitating ions are in the proper stoichiometric ratio, such as one-to-one for one-one electrolytes and two-to-one for two-one electrolytes. If the ratio is nonstoichiometric the analysis must be altered. In the following discussion we will assume that the nonprecipitating ions in solution do not form insoluble salts with the precipitating ions, so that the latter will be the only ions adsorbed in the surface layer. If the concentration ratio of precipitating ions in solution is not greatly different from the stoichiometric one the ratio of concentrations in the adsorbed layer will be stoichiometric. Thus for a one-one electrolyte one needs only to let C,, and C , in the above analysis be the concentrations of the deficient ion in solution initially and at time 1, respectively. This means that both ions are still effective in the growth process and that their concentrations in the adsorbed layer are equal because the particle remains essentially uncharged. The factor C, - Cois also equal to the concentration of either ion yet to be precipitated before ,equilibrium is reached. The appropriate Co value is now the equilibrium concentration of deficient ionj which for a one-one electrolyte can be calculated from the relation (17) R. A. Johnson and J. D. O'Rourke, J . An. Chcrn. S o c . , 76, 2124 (19.54). (18) A . E. Xielsen, J . Colloid Sci., 10,57G (1955).
PRECIPITATION KINETICSOF IONIC SALTS FROM SOLUTION
Sept., 1958 (Ce
- Cd
1071
+ CdCO = s
where C, and cd are the initial concentrations of excess and deficient ion, respectively, and 8 is the solubility product coefficient. For a two-one electrolyte the ionic ratio in the surface layer is the Ytoichiometric one, so again we need only substitute for C, and C, the appropriate molar salt concentrations corresponding to the deficient ion concentration in the solution. With these substitutions the above growth equations should apply to this case of non-stoichiometric concentration ratio. If one ion is adsorbed more strongly to the precipitate particle than the other the particle retains an electrical charge and an electrical “double layer” is set up around the particle by the ions neutralizing this charge. This situation is familiar in Colloid Chemistry.19 The particle charge tends to minimize selective adsorption, since the particle attracts the more weakly adsorbed ions of the opposite charge. Thus for ionic concentration ratios in solution equal or close to the stoichiometric one selective adsorption should be negligible, as assumed in the preceding paragraph. However if one precipitating ion is present in solution in great excesp, it may be adsorbed to a much larger extent than the deficient ion, in spite of the resultant charge. In this case the crystal growth is controlled by the concentration of deficient ion only, since the surface concentration of the excess ion remains essentially constant. Then for both models the flux equations become J = b(C, - CO) for one-one electrolytes, where C , and COare the concentration of deficient ion in solution and a t equilibrium, respectively. For a two-one electrolyteA2B,J = b(C, - Co) if ion A is in large excess, and J = b(C, - Co)2if the ion B is in large excess. In all these equations b should be constant with time and with deficient ion concentration, but it may depend upon the initial concentration of excess ion in solution, so it cannot be compared directly with b values from the previous equations. In all of the above discussion the reaction and adsorption coefficients were assumed to be invariant with concentration, although this is strictly true only if activities are used instead of concentrations. Some authors have calculated activities for interpretation of crystal-growth results from the DebyeHuckel equations, but these equations are valid only in dilute solutions, whereas the growth processes considered here all occur on the particle surface. A correct calculation of surface ionic activities is very difficult a t present, because sufficient information about the state of ions adsorbed on a crystal surface is not available. Thus only concentrations will be used in the following comparisons of experimental results with the growth equations, and the assumption of concentrationindependent coefficients will be justified from the comparisons. Unfortunately we will be unable to calculate the growth coefficient b from most of the experiniental results, since such a calculation requires knowledge of the number density of particles (or their final (19) E. J. W Verwey a n d J. Th. G. Overbecb, “Theoiy of the Stability of Lyophobic Colloids,” Else! ier, Amsterdam, 1‘348
J
TlME TIME
, MlN MIN
Fig. 1.-Turnbull’s results for the precipitation of barium sulfate; equal initial ionic concentrations of 19.1 X 10-8 moles/l. Solid lines drawn from eq. 8, dashed lines from eq. 12; m = 3.
0.9-
an-
013!
o:L!/ L A o o 06-
03 04
100
T I M E , MIN
Fig. 2.-Turnbull’s results for the precipitation of barium sulfate. equal initial ionic concentrations of 19.1 X 10-6 moles/i. Lines drawn from eq. 12, m = 3.
size). This can be seen from eq. 1; the experimental results give a W against t relation, and the required Concentrations are known, but N and Rf are not, since few workers have measured either of these quantities independently. The change of N with initial supersaturation can be found by assuming that b is constant for a given system a t one temperature, but its absolute value requires a separate measurement. Comparison with Experimental Data The equations already given above will now be compared to experimental data on the crystallization of salts from solution. Most of the experimental results were obtained by following the concentration of salt in solution by measuring the solution conductance. The value of W is then taken to be (C, - C,)/(C, - Co) where C , is the solute concentration in solution a t time t, C, is the initial solute concentration and Cois the equilibrium solute concentration calculated from the solubility product. The experiments discussed here were done at 25” unlegs otherwise noted. The results for different salts will be discussed separately. Barium sulfate has been used frequently for crystallization studies because it precipitates slowly, is very insoluble, and is important in analytical chemistry. In this discussion a value of 1.2 X niole,/cm.8 will be used for the solubility product of barium sulfate. Turnbull measured the
1072
R. H. DOREMUS
precipitation of barium sulfate by mixing equivalent amounts of barium hydroxide and sulfuric acid, so that only the precipitating ions were present in so1ution.l His precise measurements during the early stages of growth showed that W was proportional t o t 3 up t o about 5% reaction, which is the result required by eq. 8. Therefore the crystal growth was controlled by an interface process, and the number of growth nuclei was constant, as assumed in the derivations of the growth equations in this discussion. If the precipitation were controlled by bulk solute diffusion the time exponent would be 3/2, and if nucleation continued after the initial period the exponent would be greater than 3. Turnbull’s results for the precipitation of barium sulfate with an initial salt concentration of 19.1 X mole per liter are shown in Figs. 1 and 2.*O The initial concentrations were the same for the two runs shown, but the methods of mixing the solution were different. I n Fig. 1 the solid lines were drawn from a numerical integration of eq. 10 with m = 3, and the dotted lines were drawn from eq. 12 with m = 3. I n Fig. 2 the lines mere drawn fromeq. 12 withm = 3. bince m = 3 throughout both runs one can conclude that the same growth mechanism prevailed during both of them, although at a certain stage of precipitation the effective growth area became constant, rather than increasing as the particle surface area increased. The growth mechanism is that described by eq. 4, which corresponds to Model A. The number density of particles in different solutions can be compared if their b values are the same. If is defined as the ratio of t to Jo” [dz/x’/a(l - x ) ~ ] , then r is constant with W a s long as eq. 10 with m = 3 is obeyed and is given by
Vol. 62 TABLE I CALCULATION OF W ’
Salt and soln.
BaSO4,A . BaSOd, B SrS04, E SrS04, F
cm, . moles/l.
19.1 X 19.1 X 7.5 x 5 X
10-6 10-5
10-3 10-
p = 2(4rN)’/sb(Cm
7,
miZ’-1
min.
WJ
0.192 ,024 ,119 .058
4.9 13.6 9.0 14.4
0.32 .OG .39 .27
- Co)8/~(3W’)2/a/p’/a
(lG)
Therefore from data for the early and late stages of precipitation some idea of the value of W (denoted as W’) which corresponds to the onset of constant effective growth area can be found, since from eq. 14 and 16 p7 = 2(W’)%
(17)
The slopes p and W’ values for solutions A and B are shown in Table I; the W’ values are about what one would expect from Fig. 1. Collins and Leineweber measured the precipitation of barium sulfate by generating sulfate ion homogeneously in a solution of barium nitrate. Nucleation therefore was not affected by local concentration excesses, which probably initiated it in the direc t-mixing experiments. In Collins and Leinemeber’s solutions barium ion was present in considerable excess, from ratios of about 10 t o 150 to one. Their results fitted eq. 1 with Rt/D* = 0 (or eq. 10 with m = 1) throughout the precipitation process. The result of m = 1 is coneistent, with the equations found in the discussion of the effect of a large excess of one ion. Collins and Leineweber measured the size of their precipitate particles from the dissymmetry of scattered light, so it is possihle to calculate absolute values of b from their results. They chose to express their results in terms of an adsorption r = ( PI3 ) probability, as discussed above in connection with (47rN)’/ab(Cm- c o p the question of diffusion or interface control. This For two solutions 1and 2 with the same b value discussion showed that such use of an adsorption probability was improper; in fact, their result of the same adsorption probability for different solvents From this equation the number density of particles would be surprising. The value of b can be comfor solution A shown in Fig. 1 was about 22 times puted from their (Y value as follows. From their greater than N for solution B, although the initial relation for y, y = S / ( Y = D/b or b = 108Da. concentrations were the same for the two solutions. Therefore their result of a nearly constant value of I n another experiment Turnbull measured the pre- a: for precipitation in water at a constant temperacipitation in a solution (C) mixed in the same way as ture (constant D) shows that b is constant with solution B in Fig. 1, but having an initial salt con- ionic concentration in solution, as assumed in the centration of 23.7 X 10-6 mole/l. The result of derivation of the present equations. I n this case this run was a curve almost identical with that for the average value of b was about 0.024 cm./sec., solution A in Fig. 1. Thus from eq. 15 solution C which means that the initial rate of growth of the had 3.5 times the number of particles that solution particle radius was about 8 8. per second from eq. 10. R had. Turnbull tabulated the results of several Variations in the a values reported for the same different precipitations a t the same initial concen- solvent doubtless reflect the experimental untrations (see Table 11, ref. 1) and showed that their certainty in determining the particle radius from the dissynimetry of scattered light. The values of T values (particle density) depended upon the method of mixing, the solutions used and unknown N calculated from these results range from about 1.5 X lo8 to 1.5 x lo9 particles per There factors. I n the later stages of precipitation when eq. 13 is is no correlation between the supersaturation, as obeyed a plot of 1/(1 - W ) zagainst time is linear. calculated from the product of the ionic concentraThe slope p of such a line equals 2NA’b(C, - Co)2, tions at the start of precipitation, and the number density, although the number of particles roughly if A’ is computed from W ’ increases as the initial sulfate concentration in(20) The data for solution A, and for solution B in Fig. 2, have not creases. h a l l impurity particles probably were been publidied previously, and t h e a u t h o r is gratefill to nr. Turnhull the ma,jor nucleating ngcnt,s; the nuthors coiicluded for siipp1yin.g them t o lrini.
Sept., 1958
PRECIPITATION
KINETICS OF IONIC SALTS F R O M SOLUTION
that impurities affected the crystallization process from studies of precipitation with materials purified by different procedures. Assuming that the solute molecules behave as a perfect gas we can compute an adsorption probability from kihetic theory. The flux of molecules striking one square cm. of surface per second is given by C d R T / 2 n M where C is the concentration and M is the molecular weight of the molecules, R is the gas constant and T is the temperature. Thus the fraction of molecules adsorbed f is given by
4-g Cb
f=
=
for sulfate molecules. Actually, of course, the flux of molecules striking the surface probably is smaller than that given by the kinetic theory expression. Collins and Leineweber also measured the effect of solvent viscosity upon precipitation rate by crystallizing barium sulfate from various glycerolwater mixtures. Again the results should be interpreted in terms of the effect on growth coefficient rather than upon solute diffusion coefficient, since the solute diffusion in solution is not controlling the precipitation. The results showed that the product of b and the solvent viscosity was roughly constant; that is, that b was inversely proportional to solvent viscosity. This result is not surprising, since surface diffusion should be a factor in the coefficient b, and the diffusion coefficient for surface mobility is probably proportional to solvent viscosity. Johnson and O’Rourke measured the precipitation of barium sulfate from mixtures of barium chloride and sodium sulfate with different ratios of the precipitating ions.17 They attempted t o analyze their results in terms of two processes, nucleation and growth, but the results of the above two studies show that in barium sulfate precipitation nucleation is completed before the experimental mea3urements are taken, and the entire process observed is one of growth. Johnson and O’Rourke’s results for an initial sulfate-to-barium ion ratio of two-to-one are plotted in Fig. 3, using the deficient ion concentration to measure the extent of reaction. The line is drawn from an integration of eq. 10 with m = 3. The fit is good throughout the measured reaction time, showing that for this period the precipitation process proceeds by the same mechanism as for Turnbull’s experiments in the initial stages of reaction. Oden measured the precipitation of strontium sulfate at 20” when strontium hydroxide and sulfuric acid were mixed in equivalent amounts.21r22 His results for three different initial concentrations are shown in Figs. 4 and 5. Oden used a Co value of 8 X 10-4 molejl., and KohlrauschZ3 found 6.2 X mole/l. as the solubility of SrS04; a fits the data best, however, so value of 7.5 X it was used to make the plots in Figs. 4 and 5 . The solid lines in these figures were drawn from eq. 8 with m = 3, whereas the dashed lines were drawn from eq. 12 with m = 3. For the solution with the (21) S. Oden, Arkiu Kern$, Mzn., Geol., 9, No. 23 (1920). (22) 8. Oden and D. Werner, zbzd., 9, No. 32 (1926). (23) F. Kolilrariucl~,%. p k g s t k . Clrsm., 64, 121 (1008).
1079
050.5
-
O*
-
+’
0302
-
0.1
;I
Ib
do
,a TIME,d,
MIN.
A
6;
a0
,a
4k
Fig. 3.-Johnson and O’Rourke’s results for the precipitation of barium sulfate; initial concentrations: barium 12 X mole/l., sulfate 24 X 10-6 mole/l. Line drawn from eq. 8, m = 3. 0.9
1
CONC!
X IO’
’
Fig; 4.-Oden’s results for the precipitation of strontium sulfate; equal initial ionic concentrations. Solid lines drawn from eq. 8, dashed lines from eq. 12; m = 3. IO.
I
l o o
0
0
-
oii‘
CONC
x
IO’
MOLES I L
;
,p
04
1
-
P / P /
1
03 loo
TIME. MIN
1,000
10,000
Fig. 5.-Oden’s results for the precipitation of strontium sulfate; equal initial ionic concentrations. Solid line drawn from eq. 18, dashed lines from eq. 12; m = 3.
highest initial concentration eq. 10 was followed throughout the period of precipitation, but for lower initial concentrations the effective growth area on a particle apparently became fixed at a certain stage of growth. The relative number densities of particles for the three solutions can be computed with eq. 15. From this calculation solution E had 1.4 times the number of particles in solution D, and solution F had 14 times the number of particles in eolution D. These results are surprising, since the iiumher density dccrenscs with init in1 super-
1074
R. H. DOREMUS
saturation increase. It is possible that the method of mixing was not the same in each case, or that more nucleating impurities were introduced into the solutions of lower supersaturation. For the later stages of precipitation the value of W’ corresponding t o the constant area A’ can be calculated from eq. 17 as described above. These W’ values are shown in Table I, and as for the barium sulfate solutions the calculated W’ values are about what would be expected from Fig. 4. Van Hook followed the precipitation of silver chromate by measuring the silver ion concentration with an electrical cell.24 His data for three different initial silver chromate concentrations, taken from Fig. 4, ref. 24, are shown in Fig. 6. The plots of 1/(1 - W ) 3against time are linear, so eq. 12 for a two-one electrolyte ( m = 4) applies. Apparently the silver chromate crystallized by the same mechanism as barium and strontium sulfate, and the effective growth area on the crystal surface became constant early in the precipitation. Howard and Nancollas measured conductimetrically the crystallization of silver chromate from slightly supersaturated solutions aftter seed crystals were added.25 The supersaturation was so low that no nucleation of precipitate particles occurred; thus all crystallization took place on the surfaces of the seed crystals. The solutions were well stirred, so the interface reaction was controlling, although the seed crystals were larger than the particles in the experiments without added seeds. The surface area of the seed crystals changed by a negligible amount during the precipitation. The results fitted eq. 13 with m = 3, so Model B is applicable rather than Model A as found for Van Hook’s experiments. The reason for this dissimilarity could be the different bulk supersaturations in the two sets of experiments. For Model A the rate of precipitation is proportional to the fourth power of the solute concentration, whereas for Model B the rate is proportional to the third power of this concentration. Thus for high concentrations Model A should apply, and a t lower concentrations Model B should be controlling; since the solute concentrations in Van Hook’s experiments were considerably higher than in Howard and Nancollas’, this contention is consistent with the experimental results. Howard and Nancollas found that b was the same for different initial ionic ratios if the deficient ion concentration was used for C, as described in the equations section. Davies and his eo-workers have measured the precipitation of silver chloride when seed crystals were added to slightly supersaturated solutionn. 15,26 I n these experiments the seed crystals were so large (3-5 p in diameter) that their surface area did not change appreciably during the precipitation ; however, the solutions were stirred vigorously, so the interface reaction was controlling. The results fitted eq. 13 with m = 2 and N A ‘ equal to the total of solution, surface area of seed crystJals per and the growth coefficient b was the same for several different initial solute concentrations. If the (21) A. Van Hook, TH18 JOURNAL. 44, 751 (1940). (25) J. R. Howard and G. H. Nancollas, Trans. Faraday Soc.. 63, 1449 (1957). (26) C. W. Davies and G . H. Nancollar, ibzd., 61, 818 (1955).
Vol. 62
initial ionic ratio was not one eq. 13 with m = 2 was still obeyed if the deficient ion concentration was used for C,; the same value of b was found for all ionic ratios studied. Again Model B applied, apparently because the solute supersaturations were quite low. The total surface area of seed crystals N A ’was known, so b could be calculated to be about 1.5 X lo4~ m . ~ / m o l sec. e.
Discussion Let us now consider certain aspects of the growth models proposed. It would be difficult t o explain the kinetic results without invoking an adsorbed surface layer of ions. If the ions were incorporated a t the growth sites directly from solution one would expect that the rate of crystal growth would be proportional t o CF - C;P rather than (C, - Co)m as was found experimentally. I n the experiments in which the ionic ratios in solution were not stoichi’ometric the growth rates were proportional to some power of the deficient ion concentration. Thus if more of the ion in excess was added (at least in moderate amounts) the rate of precipitation was unchanged. This result could not occur if the ions were incorporated directly from solution, since then the precipitation rate should be proportional to the product of the total concentrations of both ions in solution. Therefore the experimental measurements of the kinetics of ionic crystal growth from solution are best explained if an adsorbed surface layer of ions is postulated. Fitting a solute unit directly into the crystal lattice from solution should be difficult. Even for uncharged molecules a collision at a kink in a growth step with just the right orientation to fit into the crystal lattice would be an improbable event. For ionic solutes the difficulties are even greater, because simultaneous or a t least alternate collisions of oppositely charged ions with a kink would be even more improbable events. Furthermore many ions are solvated in solution, and desolvation would have to occur a t the same time as the ion fits into the lattice. Thus the concept of an adsorbed layer, which permits desolvation and partial orientation before incorporation into the lattice, seems reasonable. I n certain of the crystallization experiments in which seed crystals were not added it was found that the effective growth area of the particle became constant a t a certain stage in the precipitation. From Figs. 1 and 4 for barium and strontium sulfates we see that this phenomenon appeared a t roughly the same time after the initial mixing in several different experiments. This result suggests that impurity adsorption influenced the formation of growth steps. The desorption of surface impurities could hinder formation of new growth steps; alternatively, impurity ions adsorbed from the solution could poison new growth steps as they are formed. Some information on the nucleation process can be obtained from the results in which seed crystals were not added. The crystallization kinetics of barium and strontium sulfates are accurately described by a growth process only, so the assumption of a constant number of growing particles for these systems is justified. Thus the numerous
Sept., 1958
PRECIPITATION KINETICS OF IONIC SALTSFROM SOLUTION
interpretations of the initial growth period in salt precipitation as an “induction period” or “time lag” for nucleation must be viewed with suspicion. In fact, nucleation during a measurable time occurs only under special conditions, such as in the experiments of Dunning and Notley on crystallization of cycloniteZ7and those of Schlichtkrull on insulin.28 Usually nucleation of crystals in solution is completed in a very short time, and nucleation rates for such a system have not yet been measured experimentally. The above interpretations show that the differences in lengths of initial periods of growth are due to differences in both the number of nuclei present aiid the growth rate. The growth rate is strongly affected by the initial supersaturation, shown by the factor (C, - Co)m in eq. 10, and this effect is the one generally attributed to different nucleation rates by authors who theorize about the “induction period.” The influence of initial supersaturation upon the number of nuclei in the crystallizing salt solutions discussed above was less than generally supposed. Turnbull’s results on barium sulfate showed that the method of mixing the solution could change the number density of particles by a greater factor than substantial variations in supersaturation changed it. Davis aiid Jones’ experiments on silver chloride also showed a large variation in growth rate when solutions of the same initial solute concentrations were mixed in different ways.29 The experiments of Collins and Leineweber on barium sulfate precipitation showed no consistent effect of initial supersaturation on particle number density, even though the nucleation process occurred uniformly throughout the solution, while Oden’s results on strontium sulfate showed a decrease of particle density with increase of supersaturation, contrary to the expected increase. Thus the number of nuclei in these salt solutions was not entirely dependent upon solute supersaturation. The nucleation was doubtless heterogeneous, and the probable nucleating agents were minute colloidal impurity particles. The particle number density in Collins arid Leineweber’s experiments varied from 1.5 to 15 )( lo8particle~,/cm.~, which is within the range of possible impurity particle density. A spectrum of nucleating probabilities aiid different concentrations for the impurity particles could account for the variations in the density of precipitate particles with method of mixing, supersaturation and other uii kn own factors. In conclusion the main results of this discussion are as follows. 1. For the systems considered the crystal growth was controlled by a process a t the particle surface, rather than bulk diffusion of solute. 2. If no seed crystals were added nucleation occurred very rapidly when the initial solutions were mixed, and the number of salt particles remained constant throughout the measured growth (27) W. J. Dunning and N. T. Notley, Z.Elekt., 61, 55 (1957). (28) J. Schlichtkriill. Acta Chem. Scand., 11, 439 (1957). (29) C . W. Daiies and -4. L. Jones, Dzsc. F m a d a y Soc., 6, 103 (1949).
m
,
,
,
,
,
,
,
,
I
I
1 -
0
I5
;o
25
& TIME
35 MIN
1075
” 1
A
Fig. 6.-Van Hook’s results for the precipitation of silver chromate; equal initial ionic concentrations. Ordinate scale must be divided by ten for solution I.
period. 3. Surface adsorption of ions was found to be a necessary step in the growth process. 4.Functional relations between growth rate and solute concentration were established for barium and strontium sulfates. 5. These relations and those for other salts were rationalized in terms of two models for the incorporation of solute into the crystal lattice. 6. Factors such as the density of impurity particles and the method of mixing solutions were as important in the nucleation process as solute supersaturation. Acknowledgments.-The author wishes to thank D. Turnbull for introducing him to this subject and for making suggestions on the manuscript. He is also indebted to G. W. Sears, D. A. Vermilyea, F. S. Ham and H. W. Cahii for etimulating discussions. Appendix Simplified Derivation of Eq. 1.-From Fick’s first law of diffusion J = DG in which J is the flux of diffusing species per unit area and G is the concentration gradient of diffusing species. For threedimensional growth of an isolated spherical particle from dilute solution Zener4 finds that G = ( C , C,)/R a t the particle surface, where C, is the solute concentration a long distance from the particle. To account for competing particles in a dilute solution C, can be replaced by (C, - Co)(l Co. When these relations are combined with that for the flux J = b(C, - C,) in the interface reaction and solved for C, the result is
w)
+
D(Cm - Co)(l - W ) D bR
+
+ GI
Since t,he deposition of solute on the particle surface is assumed to be uniform
The solution of this equation for particles of negligible initial radius is eq. 1.
