THE JOURNAL OF
PHYSICAL CHEMISTRY (Registered in
VOLUME62
U. 8. Patent Office).
(0Copyright, 1968, by the American Chemical Sooiety)
SEPTEMBER 24, 1958
NUMBER 9
PRECIPITATION FROM SUPERSATURATED SOLUTIONS OF INSOLUBLE AND SPARINGLY SOLUBLE METAL SALTS. PART I. THE PARTICLE SIZE OF CRYSTALLINE PRECIPITATES BY A. PACKTER 106 Howbemy Road, Middlesex, England Received June BO, 1066
The factors that determine the crystal size of precipitates from supersaturated insoluble metal salt solutions a t different supersaturation are discussed. The experimental data are best ex lained by a two-stage consecutive nucleation and condensation mechanism for precipitation, rather than by a sol coagugtion mechanism. Duke and Brown’s analysis for such a former process has been extended to take into account those factors other than interfacial energy, that may determine the rate of the nucleation rocem. At any supersaturation, the or stal size of precipitates of salts of widely different solubilities is mainly determined y ! ! the solute concentration, rather than g y the rate constants for nucleation or crystal growth.
C
I
Introduction The properties of insoluble metal salt precipitates from supersaturated aqueous solution have been studied by von Weimam, OdBii, Kolthoff and Balaand more recently by TeiakS6 The factors that determine the stability of such solutionsLe., the induction period that generally precedes precipitation-also have been investigated very extensively; and Nielsen recently has reviewed all the data on this subject. Two main theories have been developed to explain the precipitation of insoluble metal salts. Balarev4 has suggested that growth of the precipitate crystals occurs by the coalescence of “submicrons” of colloidal size present in the supersaturated solution, by a coagulation mechanism; and many other workers, although less explicitly, have also assumed a colloidal nature for supersaturated metal salt solutions. 1,3s6-7 On the (1) (a) P. P. von Weimam, Chem. Revs., 2, 71 (1926); (b) KolEoid Z., 2, 199, 230, 275 (1908); (0) 3, 282 (1909); (d) 42, 305 (1927). (2) 8. O d h , Arlciu. Kemi. Min. GeoE., 9, 25 (1926). (3) (a) I. M. Kolthoff. et al., J . Am. Chem. Soc.. 56, 1264. 1658 ( 1934); 57, 597 (1935); 68, 121 (1936); 60, 39, 197. 499, 505, 508 (1938). (4) (a) D. I. Balarev, Kolloid-Beih.. 50, 1 (1939), (b) C. A . , 49, 129lOef (1955). (5) (a) B. TeZak. et a!.. 2. phyaik. Chem., Al75, 284 (1936); (b) A190, 257 (1942); (c) THISJOURNAL, 67, 301 (1953); (d) J . Colloid Sci. (Suppl.), 18 (1954). (6) A. Packter and R. Matalon, Disc. Favaday Soc.. 18, 161 (1955). (7) (a) E. N. Gapon, J . Russ. Phyr. Chem., 61, 512 (1929); (b) E. Pozner. J . Phys. Chem. U.S.S.R., 13, 889 (1939); (c) W. Takami. Kagaku. 2 8 , 428 (1953); (d) F. Brescia, et al., J . A m . CAem. Soc., 7 6 , 9946 (1954).
basis of this theory, the final crystal size would depend mainly on the degree of aggregation or “polymerization,” which in turn would be determined by the rate of coagulation of the sub micron^.^^^ On the other hand, van Hook,’Oa La Mer’ob and Dunningloc have proposed that the crystal growth takes place by condensation of the ions of the metal salt onto small nuclei that are first formed in the supersaturated solution. The rate determining step for nucleation is the formation of a “cluster” or “embryo” that consists of a small number of reacting ions. The kinetics of such a consecutive and autocatalytic reaction have been analyzed by several workers,11-16who showed that the average crystal size of the precipitate depends on the relative rates of the nucleation and growth processes, and as some high power of the concentration of reacting ions. The rate of nucleation in turn varies with the supersaturation according to the VolmerWeber theory. (8) H. Dorstal, Monats.. TO, 334 (1937). (9) F. C. Collins and G. E. Kiniball. J . Colloid Sci.. 4, 425 (1949). (10) (a) A. van Hook, THISJOURNAL, 45, 422, 879, 1194 (1941); (b) V. La Mer and R. H. Dinegar. J . A m . Chem. SOC.,7 6 , 2124 (1951); (c) W. J. Dunning, Disc. Faraday Soc., 5 , 79 (1949). (11) F. R. Duke and L. M. Brown, J . A m . Chem. SOC.,7 6 , 1443 (1954). (12) R. A. Johnson and J. D. O’Rourke, ibid., 76, 2124 (1954). (13) (a) 0. M. Todea, Acta Phusicochem. U.S.S.R., 13, 617 (1940): (b) Problem Kinetiki, Akad. Nouk, S.S.S.R., 7 , 91 (1949); C. A . , 48, 133233 (1954). (14) M. Roginskii, Acta Physicochem. U.S.S.R., 10, 825 (1939). (15) 8. H. Bransom and W. J. Dunning, Disc. Faraday SOC.,I, 96 (1949).
1025
1026
'
A. PACKTER
Although the particle size of a "crystalline" precipitate can be determined accurately by simple techniques over a far wider range of concentrations than can induction period, less quantitative experimental data had been presented on this property. (Von Weimarn studied only barium sulfate, lead halides and a few insoluble silver salts.) However, Duke and Brownll recently have extended these studies to metal chelates, JonkerslBand van Schulenburgh17 have discussed the properties of microcrystalline silver bromide and ferric hydroxide precipitated a t very low supersaturation, whiIe the author1*has presented new data on a series of crystalline "peptized" insoluble silver, copper and lead salts. It is therefore now possible to test these two theories with data on the variation of crystal size with supersaturation of precipitating solution for metal salts of a wide range of solubility. Earlier studies on stability of supersaturated solutions are also rather unsatisfactory from a theoretical viewpoint, since most workers (including the author) have estimated rates of nucleation and of coagulation as the reciprocal of induction period~,~-~OaJO~.~~ without any theoretical basis for this assumption. Johnson and O'Rourke, l 2 however, have recently proved that stability depends on the rates of both nucleation and crystal growth; and it would thus be valuable to re-examine earlier data in the light of their theory, This series of papers presents such a quantitative analysis of previous studies on crystal size and supersaturated solution stability. Earlier work on the effect of additives also will be re-evaluated. I n the first paper we have studied the variation of particle size of crystalline precipitates with supersaturation. The theoretical analyses of Duke and Brown for a consecutive two-stage condensation process for precipitation has been developed further to include those factors, other than surface energy, that may affect the rate of the nucleation process: a semi-quantitative analysis of a precipitation process that occurs according to Balarev's theory also has been presented. The experimental data are best explained by the former theory. Theoretical We first discuss the relations between crystal number (and size) of precipitates from supersaturated solutions, and the absolute solute concentration, predicted by the two main theories proposed by previous workers. I. Growth by Nucleation and Condensation onto Nuclei. (a) Number of Crystals.-Duke and Brownll have noted that the number N , of precipitate crystals deposited from supersaturated metal salt solutions rises rapidly with the solute concentration (C) a t intermediate and high supersaturations. For precipitation from equivalent solutions prepared by rapid mixing of the reactants, they have proposed the equation
Vol. 62
kl and kz are the rate constants for nucleation and growth onto nuclei, K is a function of M and p (molecular weight and specific gravity of the solute) ; m and m' are the orders of reaction in the nucleation and growth processes. These workers, however, did not consider the effect of decreasing rates of nucleation (predicted by the Volmer-Becker theory on crystal growth at low supersaturation). We have therefore extended their analysis to take this and other factors into account. (i) Duke and Brown assumed that the rate of nucleation a t equivalent ion concentration C, dN/dt = k 1 C m since nucleation is determined by supersaturation rather than by absolute ion conwe propose that the equation dN/dt = klsm, would be more appropriate in the above analysis, where s is the supersaturation constant. [s = (C - Cs)/Cs, where C, is the saturation solubility in distilled water.] Similarly, for the rate of crystal growth, we would substitute the relation - dN,/dt = kza(s 1)m'. C,"' in place of - dN/dt = kzaCm', where a's are the total surface area of the nuclei. Equation 1 then will become
+
+
(ii) Recent work10b~20 on the kinetics of crystal growth onto seeds of insoluble metal salts has confirmed that m =2; i.e., (2) becomes N8
=
+
Kkis" particles/l. kzCs2(s 1)
(3)
For larger values of s, (3) then becomes N,
=
KklSm--2
2particles/l. k2G
(iii) Earlier workerslOc had considered that the value of kl was determined by the interfacial energy per unit surface area of the nucleus. Recent studies have indicated that more generally16J1-23
where $1 is a constant that depends 011 the physical ~ the ~ )energy ~ of actiproperties of the salt, ( L L is vation required to overcome the interfacial forces, and (AA1)non. is the energy of activation required to overcome electrostatic and steric effects. (In unstirred systems the (AAl)nonr term will also inelude a diffusion term (AA)y;sc.)24(AA1) in turn equals TU?%', and varies with the supersaturation (s), while (AAl)nonuis independent of the supersaturation. (a represents the solid-liquid surface energy per unit surface energy of the nucleus, rk is the critical radius of the nucleus; and y is a geometrical constant depending on the shape of the nucleus.) rk in turn varies with s, according to the relation (20) C. W. Davies and A. L. Jones, TTana. Faraday Soc., 41, 802
(16) (17) (18) (19)
G . H. Jonkers and H. R. Kruyt, ibid., 18, 170 (1955). J. yan Schulenburgh, et al., Rec. trau. chim., 68, 999 (1949). A. Packter, Z.physik. Chem., 207, 210 (1957). A . E. Nielsen, J . Colloid Sci., 11, 520 (1955).
r.
+
(1955). . .
*
(21) B. 8. Srikantan, J . Ind. Chem. SOC.,SO, 469 (1950). (22) K. Hirano, Busssiron i K e n k y o F u k u s h i m a , 5, 100 (1953); C. A . , 47, 3667i (1953). (23) R. Gopal, J . I n d . Chem. SOC.,20, 62 (24) A. van Hook and A. J. Bruno, Disc. Faraday SOC.,5, 112 (1949).
(1943).
c
,
,&
#I
PARTICLE SIZEOF CRYSTALLINE PRECIPITATES
Sept., 1958
1027
(5)
where M,p are the molecular weight and specific gravity of the metal salt, and 0 is the absolute temperature of the supersaturated solution. rk falls rapidly from infinity a t very low supersaturations t o a limiting value of T k - 1 0 A. at high s u p e r ~ a t u r a t i o n s , ~and ~ ~ ~(AA1) 5 u varies accordingly. kl also falls rapidly with s at low s values, but we may note that a t high s values, kl will approach some asymptotic value, according to the relation (log k1)8+rn
=
,451
-
(AAdnonc ~
RO
(6)
Similarly, k2 varies with (AAz) , the activation energy for crystal growth onto nuclei, according t o the relation where dzis another constant. log kz = 62
( A 4 -Re
(7)
T o take into account the variation of ICl, with both (AAl)u and (AAl)n04u, and the variation of the former with s, equation 2 is best expressed in a general logarithmic form (to the base 10)
Fig. 1.
from a supersaturated solution according to a nucleation and condensation process has been analyzed by several workers.20.21,23J5For “autocatalytic” growth where no appreciable number of new nuclei are formed after the induction period, the size distribution will be “left-skewed”: and with fall in At high s values, this relation becomes kl/kzCs2 values, the mode value of d3 may rise from about 0.5 to >0.9 damax, where &ax is the diameter of the largest crystal. The mean deviation of the final crystal crop corresponds t o that of the original nuclei, on which these crystals have In Fig. 1, curve I represents graphically the varia- grown, and will thus decrease with decrease in kl/kZCS. tion of (AA1)u/R0 with s, according to the equation (11) Growth by Coagulation.-The shape of the rate of growth-time curves for precipitation of insoluble metal salts would indicate that a proand curve I1 represents the variation of the function longed “induction period” is followed by a rapid utilization of m i c e l l e ~ . ~ ~ l ~Therefore, bJ~ if precipitation does occur by a coagulation process, this Le., after the inducmust be largely autocatalyti~,~~ with s1 according to the equation tion period, coagulation must occur predominantly by growth (of micelles) onto the aggregations of largest surface area that have already formed. Curve 111, the sum of I and 11, then represents (a) Number of Crystals.-We have applied graphically the variation of the function log N s = Duke and Brown’s method of analysis to such an R(s) = P(s) Q(s) with s, according to equation autocatalytic coagulation. The rate of nucleation 8. N, the extrapolated value of Q(s) a t s = 1, now varies as klamC2where a, is the average surface may be expressed by the relation area of the sol micelles, and kz now equals kl; whence equation 3 becomes K - [ ( A A ~ ) n o n c- ( A A d I (9)
+
log N ’ = bz
+ log
Re
(b) Size and Size Distribution.-The modal average diameter, dmode, varies with N,, according t o the simple relation where d represents the diameter of the cube of volume equivalent to that of the crystal.21 The size distribution of a precipitate deposited (25) J. Harbury, THIS JOURNAL,61, 382 (1947).
More exactly, substituting in the Duke and Brown analysis, the rate of formation of new nuclei per unit weight of precipitate, dN/dw now equals A (cf. ref l l ) , and dN/dw = a,/A = M 1 /p2/‘am S w,,/s (where W is the weight of nuclei). (26) E. F. Burton, “The Physical Properties of Colloidal Solutions,” Longmans, Green B: Co., London, 1938, Ch. 12,p. 210.
A. PACKTER
1028
N ‘ / d N = (~.,(p/M)~/aW-~/adW N‘/rdN = am(p/M)%Wo% (12)
ie N =
(p/M)’/zW’/4cyma/4
- a,3/4p‘/2cg=/4s1/4 M,/,
particlesjl.
(13)
The number of crystals would here thus be inde, vary only to a small extent with pendent of k ~and metal salt concentration, but would depend on (UJ/4.
Generally, for any coagulation process mainly autocatalytic, we might therefore expect that N will vary with C, according t o a relation N, = constant X s@‘,where p’< 1. (b) Size and Size Distribution.-Autocatalytic slow coagulation will lead to the same type of leftskewed distribution expected for growth by condensation. The mean particle size deviation will correspond to that of the aggregations onto which the rapid growth has taken place. We may note that coagulation theories do not include any explanation of the very high stabilities of insoluble metal salt systems at low supersaturation. Discussion of Published Data We have next tested the theoretical equations derived, with the available experimental data for variation of crystal size and size distribution (of sparingly soluble salts) with supersaturation, from papers by von Weimarn, Duke and Brown, and the author. The experimental methods used to determine crystal size, etc., were essentially the same in all cases. (a) Crystal Size and Number.-For a supersaturated solution of metal salt concentration, C, the final crystal size d,, is determined by the number N,, of crystals, according to equation 10.
According to either equations 1, 8 or 13, N , should in turn vary with s, at higher supersaturations, according to a relation of the general form, N , = constant X sp. We have therefore studied the variation of d, and N , with s, for a series of silver, copper, nickel and lead salts, by plotting log N , against log s. Some typical graphs have been presented in recent work.llJs These log NE-log s graphs in all cases show the general form proposed in Fig. 1 for an autocatalytic condensation process; and consist of two or in some cases three portions. A rapid rise in N , with s a t low s values along AB is followed by a straight line portion BC; and this in turn may be followed by a further section CD, along which the increase of N , with s gradually falls off. As we have demonstrated above, AB represents the rapid rise of kl, and thence N,, with s a t low supersaturations according to the Volmer-Becker theory. As the critical nucleus radius approaches its minimum value in the range of higher supersaturations represented by BC, N , then varies with s according t o the relation log N , = log Nz p log s (14)
+
Vol. 62
where N Zis the extrapolated value of N , at s = 1. The inflection a t B is generally observed at Y values of 2 to 10. For metal salts of molecular weights within the range 200 to 300, and of specific gravity of 3 to 5, these values would correspond to u values of the order of 12 to 50 ergs/cm.2; i e . , rk values of the order of 8 to 20 A. We have found that p > 3 for all the systems examined. This fact is further evidence for the condensation character of the precipitation process; since a coagulation mechanism would require that p< 1. (b) Crystal Size Distribution. Technique.Size distribution of a precipitate is generally expressed in terms of the standard deviation for weight (D), and of the maximum weight function cfn); while orderliness of growth is represented by the skewness (Am) of the crystal weight distribution curve. 15.19,27.28 The “skewness” of the crystal weight distribution curve Am is positive for all the systems examined up to high supersaturations; and the crystal weight distribution is of the type predicted for an “autocatalytic” growth process. For a particular salt, the size distribution generally becomes narrower with increasing supersaturation and reduced number of crystals. However, whereas D < 0.2 for precipitates of the very insoluble salts such as barium sulfate even at high N (ie., kl/kL’s2) values, the precipitates of more soluble salts such as lead iodide, silver benzoate, copper pyridine thiocyanate, etc., show far higher D values at similar N values.18 The far wider size distribution about the mode in such systems indicates that growth to “crystalline” size has occurred on nuclei of a correspondingly wide size distribution. (c) The Energies of Activation for Nucleation and Crystal Growth.-It would be of interest to compare the (AA1)?,,: ,. and AAz values of series of metal salts, but individual values of these activation energies cannot be obtained from crystal size data alone. However, kl/kz ratios for different salts in solution of a high supersaturation depend on the functionfi(A) = ( A A J n o n - AAz; and we might expect that this function would vary significantly with the solubility of the metal ~ a l t . ~ J Such ~J~ variations would be of some theoretical interest. We have therefore subdivided these simple sparingly soluble metal salts, whose precipitation behavior has been quantitatively described to date into various series according to their chemical characteristics, and have compared the f l ( A ) values of salts in each series with solubility ((7,). Technique. Comparison of fl(A) Values.-It follows from equation 9 that if as a first approximation the constant dz is considered constant for any series of metal salts of similar molecular weight and specific gravity, then the values of fl(A) of the members of such a series can be compared by comparing values of the function. Log N’Cs2. N ’ the extrapolated value of &(s) a t s = 1 (refer to Fig. 1) may in turn be estimated from log N, - s curves. In Fig. 1 we note that as s -+ 0 3 , P(s) + log N I : (27) A. Paokter. T ~ r JOVRNAL, s 69, 1140 (1955). (28) J. Turnbull, Acto Metalluruica, 1, 684 (1953).
r
I
.& n
I
PARTICLE SIZEOF CRYSTALLINE PRECIPITATES
Sept., 1958
and to a first approximation, for values of s greater than s (the point of inflection of the curve 111 (ABC)) R ( s ) = &(s) log Ni i.e. Q(s) = R(s) fog Ni (15) The line BC extrapolated back cuts OY a t p’; and
+
-
log N P = R. = 1
(16)
Therefore
Nz - log NI C, values have been taken from the literature. We have reported our results in earlier work. Although the values of C, in a particular metal salt series may vary by a factor of lozo,neither the values of fl(A), nor of f&4) = fl(A) - log C?, show such a wide variation: N . values at any supersaturation have not varied by a factor of more than lo3 to lo6 for crystalline salts of widely different physical and chemical properties. I n other words, although (AA1)non, AA2 and C, values may individually vary appreciably from salt t o salt, the over-all values of fl(A) and f2(A) show far smaller variations. Discussion. General Two main concepts have been proposed to explain the precipitation of sparingly soluble metal salts from supersaturated solution, the nucleation theory and the coagulation theory. The detailed kinetics for a precipitation process that occurs according to Balarev’s4 theory have still t o be completely developed; but it is clear from the growthtime curves for all the systems examined to date, that should precipitation of insoluble metal salts occur by some slow coagulation process, then this must be autocatalytic in character. The rate of coagulation would hence depend a t any instant on the surface area of the growing aggregates, and the crystal number in such systems would only vary to a small extent with supersaturation over a wide range of solute concentrations. This has not been noted. Furthermore, the coagulation theory does not include any explanation for the exceptionally high stability of metal salt systems a t very low supersaturation-or for the large size of crystals deposited from systems of the “slightly soluble” metal salts that show no specific colloidal characteristics. The experimental data discussed in this work are indeed best explained by the extension of Duke and Brown’s “theoretical analysis” for an autocatalytic condensation growth onto nuclei. There is no sharp differentiation in the kinetics of precipitation between colloidal salts, “peptized” crystalline salts of low solubility, and those crystalline salts such as lead ‘chloride and potassium chlorate, the solubility of which approaches that of the socalled “soluble” salts. A complete analysis of precipitation of sparingly soluble metal salts must, log N ’ = Qs-1 =
Ra-1
- log N1
=i
log
1029
however, take into account colloidal phenomena, since at all supersaturations, the growing particles a t some stage pass through the 100 to 1000 A. size range. Even although their adsorbed charge may only stabilize this phase transiently, such a “peptization” may modify the kinetics of the precipitation. One phenomenon that requires explanation, for example, is the narrow size distribution noted in barium sulfate precipitate~l~; since it would appear from a careful a n a l y ~ i s that ~ ~ ~this ~l~ salt ~~~ is more the exception than the rule in this effect. Precipitates of larger average particle size show a wide size distribution that corresponds to the wide size distribution of the nuclei onto which crystal growth has occurred; and narrow size distribution must therefore be due to the absence of smaller particles of colloidal size amongst such nuclei. A suitable explanation of the adsorption phenomena associated with “peptized” ~rystalline3.~J~bJ8.29 salts is also still lacking. The number (Ns) of crystals deposited from unit volume of a supersaturated metal salt solution varies with the supersaturation constant (s), according to equation 3. (Refer t o Figs. 1 and 2.) The very rapid rise in N , with s at low supersaturations corresponds t o the increase in rate of nucleation with decreasing cuticle nucleus radius Tk predicted by the Volmer-Becker theory.l°C At higher supersaturations, rk falls to its minimum value, (A&) u becomes progressively less significant and the rate of nucleation depends mainly on the term (AA1),,,,. N , then varies with s according to the mass-action equations 10 and 14. The individual values of the activation energies (AAl), (AAl)nono and AAz cannot be determined directly from crystal size data30; but the function fl(A) = [(AA1)no,, AAz] may be estimated. I n the series of simple crystalline metal salts that have been studied to date, however, we have not noted any appreciable variation of this function with solubility (C). It follows then from our analysis that the main factor that determines crystal size and number a t any supersaturation is the actual value of (l/Csz), rather than that of Icl/k2, the ratio of the rate coefficient of nucleation t o that of crystal growth. I n supersaturated solutions of very insoluble salts, the number of nuclei formed immediately ‘after mixing is generally so great that the solution is rapidly exhausted of solute. Further growth onto the nuclei is limited, and only results in particles of colloidal size. I n solutions of salts of higher solubility, on the other hand, even if k2 is small, the high value of the (1/Cs2) factor will lead to rapid growth onto the smallest nuclei, immediately they are formed. This effect may rapidly deplete the solution of solute, and a small number of larger crystals of size greater than 1 micron is generally deposited.
-
(20) J. Herak, et al., Arkiv. Kern. (Zagreb), 27, 117 (1956).
(30) T h e determination of activation energies from crystal size and induction period data will be the subject of a later paper.
