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Kinetics of crystal growth of calcium oxalate monohydrate
Journal of Crystal
Growth 21 (1974) 267-216 0 North-Holland
KINETICS OF CRYSTAL
G. H. NANCOLLAS State
University
Publishing
Co.
GROWTH OF CALCIUM
OXALATE
MONOHYDRATE
and G. L. GARDNER
of New York at Buffalo, Chemistry
Department,
Buflalo, New York 14214, U.S.A.
Received 24 September 1973; revised manuscript received 12 November
1973
The growth of calcium oxalate monohydrate crystals in stirred suspensions has been studied by following the changes in ionic conductivity in supersaturated solutions containing both stoichiometric and nonstoichiometric concentrations of lattice ions at various temperatures in the range 1545 “C. The reaction can be represented in all cases by a kinetic equation which involves a quadratic dependence of the rate of crystallization upon the relative supersaturation. This points to a process in which the crystal growth is controlled by a reaction at the crystal surface rather than by diffusion of lattice ions in the bulk solution. The effect of several inorganic phosphates, polyelectrolytes and organic dyes on the growth kinetics has been studied and the heat of solution of calcium oxalate, determined by a direct calorimetric method, 5.33 kcal mole-‘, may be compared with the value, 5.5 kcal mole-‘, calculated from the temperature coefficient of the solubility product. The thermodynamic association constant for the 1 :l calcium oxalate complex at 25 “C, determined by the conductimetric method, is 1537 1 mole-’ and there is no evidence for higher order species under the conditions of the crystallization experiments. The kinetic results are discussed in terms of the important problems relating to the origin and growth of urinary calculi.
1. Introduction The importance of calcium oxalate formation in solution in both analytical and biological fields points to the need for a quantitative study of the ionic interactions in solutions of this electrolyte and also the kinetics of its crystal growth. The crystallographic structurelp3), morphology4Y5), and composition436’7) of urinary stone materials have shown conclusively the presence of calcium oxalate in nearly all kidney and bladder stones. The calcium oxalate is either the only inorganic material involved or forms a major component of the stone together with other phosphate and urate compounds. The occurrence of various crystalline compounds within stone materials indicates the probable importance of epitaxial growth both at the nucleation and later growth stages of the calculi formation. Such epitaxial relationships between calcium oxalate monohydrate (whewellite), calcium oxalate dihydrate (weddellite) and apatite have recently been discussed by Lonsdale**9). Clearly, in addition to the growth kinetics, the inhibition of stone formation is another area of active interest since in normal urine the product of the ionic calcium and oxalate concentrations is more than one hundred times the value found in saturated solutions of the salt. Various studies”-“) have been carried out recently to elucidate this protective mech-
anism by investigating various types of compounds and ions which are effective in preventing or retarding the growth of calculi. Alexandrev, Crematy, and Crawfordl’) in a recent article have studied the effects of a number of polyelectrolytes on the crystallization of calcium oxalate and conclude that certain urinary polymers adsorb on crystal embryoes and on the faces of the growing crystals, thereby reducing the nucleation and the number of growth sites. Previous studies13-’ “) concerned with the growth of calcium oxalate have dealt with spontaneous precipitation which involves both nucleation and crystal growth. Since both processes may be occurring simultaneously, the kinetics of the reactions are difficult to interpret and the results are very irreproducible. In addition, there is an added factor which was not taken into account by these authors since various hydrates such as CaC204 - H,O, CaC,O, * 2Hz0, or a mixture of the two will nucleate out of solution depending upon the initial supersaturation, temperature and calcium to oxalate molar ratio. The kinetics of nucleation and growth of these individual hydrates may be quite different. The present study is concerned with the kinetics of growth of the monohydrate crystals of calcium oxalate from supersaturated solutions. These solutions are inoculated with well-characterized seed crystals and the 267
26%
G.
H.
NANCOLLAS
rate of crystal growth is followed conductimetrically. This method enables the accurate determination of relative changes of ionic concentrations as small as 0.02 %. The effect of changes of temperature, fluid dynamics, calcium to oxalate molar ratio and the presence of surface active materials on the growth rate have also been investigated. In addition, preliminary experiments have been made to study the epitaxial growth of calcium oxalate on seed crystals of hydroxyapatite (hereafter, HAP; Ca,,(PO,),(OH),). 2. Experimental Solutions were prepared by weight using reagent grade chemicals that had been recrystallized once from conductivity water. Calcium analyses were made using the EDTA titration method with murexide as the indicator and potassium oxalate solutions were analyzed by exchanging the alkali metal ion for hydrogen ion on a Dowex 50 ion-exchange resin and titrating the liberated acid with standard base. Seed crystals, “A”, of calcium oxalate monohydrate were prepared by the drop-wise addition of 100 ml of 0.420 M calcium chloride and 100 ml of 0.426 M potassium oxalate to 600 ml of conductivity water maintained at a temperature of 75 “C. The reaction mixture was stirred continuously during the addition which required about 2 hr; the pH was adjusted to approximately 6.20. The seed crystals were digested at 75 “C for 5 hr, washed several times by decantation until free from chloride, and were aged for about 3 months at 25 “C before use. Dilute seed suspensions, A-l, A-2, A-3 and A-4, containing 1.21, 6.0, 11.5 and 0.60 mg of crystals per ml, respectively, were used for growth experiments. Seed crystals “B” were prepared by slowly adding, with stirring, a 0.25 M oxalic acid solution to a suspension of 20 g of calcium carbonate at 85 “C. The acid was added until the pH of the seed suspension remained at 4.00; the crystals were digested for 8 hr and then left overnight in the acid solution. Crystals prepared using the latter procedure were washed with water until the pH of the seed suspension was about 6.50; dilute suspensions B-l and B-2 contained 68.0 and 10.0 mg of crystals per ml, respectively. Crystals from suspensions “A” and “B” viewed under an optical microscope (Unitron Series N-No. 54372), appeared prismatic, often as aggregates, with approximate edge lengths of l-2 urn and 4-5 urn respectively. Replicas viewed under the electron microscope
AND
G.
L.
GARDNER
showed fibrous crystals with some aggregation. X-ray powder diffraction photographs were similar to those reported1 7’18) for CaC,O, . H,O with no evidence for the dihydrate form. Supersaturated solutions for growth experiments were prepared by the slow mixing of solutions of calcium chloride (or calcium nitrate) and sodium oxalate (or potassium oxalate) in a thermostated conductivity cell. The latter, of 300 ml capacity, was of the Hartley and Barrett type”) constructed from Pyrex glass. Carbon dioxide was excluded from the air space above the cell solution by a slow stream of nitrogen presaturated with conductivity water. Stirring was achieved using either a variable speed magnetic stirrer or a vibrating glass disc (Vibromix Model E). Crystal growth was initiated in these stable supersaturated solutions by the rapid addition of aged seed crystals and resistance measurements were made using a Jones and Joseph screened ac bridge*“*i) or a transformer ratio arm bridge (Wayne Kerr Universal Bridge B221). Precautions normally taken for high precision conductivity measurements were followed. In experiments where additives were used, the same procedure was followed and the freshly prepared additive solutions were added to the cell prior to the introduction of the seed crystals. Experiments involving the growth of calcium oxalate on seed crystals of HAP were carried out at 25 “C using solutions of initial supersaturation in the range 2.0 to 3.0 containing equal concentrations of ionic calcium and oxalate ions. After approximately 30 minutes these stable supersaturated solutions of calcium oxalate were inoculated with seed crystals of HAP, prepared and characterized as described previously*‘), and the reaction was monitored following the change in solution resistance over a 24 to 48 hr period. The differential calorimeter used for studying the heat of precipitation of calcium oxalate was similar to those used previously 23,24). The sensors were 100000 ohm thermistors (Veto No. A189, Victory Engineering Corp.) in a dc Wheatstone bridge arrangement. The calibration heaters (19.591 and 19.685 ohm) were wound from enameled No. 30 Evanohm wire in an open helical form. A typical experiment involved placing 4.0 ml of a known concentration of calcium chloride or sodium oxalate in the mixing device of the calorimeter and 304 ml of a solution of either sodium
KINETICS
OF CRYSTAL
GROWTH
OF CALCIUM
oxalate or calcium chloride in the Dewar flask. After equilibration overnight (25 f 0.005 “C), the precipitation reaction was initiated by opening the mixing device and the corresponding temperature change was recorded; the reaction was effectively complete within 1 min. Two or three electrical calibrations were made before and after each experiment, the current being adjusted so that the corresponding recorder deflection for the electrical calibration was approximately the same as that anticipated for the precipitation reaction. A heat of dilution experiment, into water, was performed simultaneously in each case in the second Dewar of the differential calorimeter; temperature changes of the order 2.5 x lo-’ “C were measured with a sensitivity of approximately 1.O x 10e4 “C. The extent of ion association in solutions of calcium oxalate was calculated from the steady resistance readings in the supersaturated and subsaturated solutions of the salt. In addition, further experiments were made in which weight buret additions of either calcium nitrate or potassium oxalate were made into a weighed amount of conductivity water. The other reagent was then added in small amounts and the resistance recorded. All numerical calculations were made using either an IBM 7044 or a CDC 6400 computer.
In order to be able to analyse the results of the crystal growth experiments, it is necessary to take into account the association of calcium and oxalate ions in reaction (1) : Ca& +
C204& * CaC204(,,,.
The thermodynamic K=
association
CCaC2041 [Ca”]
[C,O:-]
(1) constant
is given by:
1 f?
where fi is the activity coefficient of a divalent ion. The measured specific conductivity, x, of solutions containing mixtures of calcium nitrate (Tc,, total molar concentration) and potassium oxalate (Tc20J can be written IO3 x (observed) +2Tc,
&(No3)2
If no ion
= 2TcZo4 AK2cZo4
-Wa~2~41 &aC204,
where K is the specific and A, the equivalent tance.
(3) conduc-
MONOHYDRATE
association
specific conductance lo3 x (calculated)
took
269
place,
the
calculated
would be given by = 2TcZo4 A KS204
+=Ca
fka(No~jr
(4) The concentration [CaC,O,]
of complex
is therefore
= IO3 [x (calculated-x
(5)
Equivalent conductivity values (A) were using appropriate Onsager equations’ “) A Ca(NOs)z = A CaCl* = A NazCz04 = A KzCz0.1 = A CaC204 =
given by:
(observed)]/2
+A CaC20,.
130.95- 143.7 135X5- 145.3 124.25 - 145.0 147.65 - 152.1 133.65-243.1
calculated
I+, I+, I+, I+, If.
Values of [CaC,O,] were calculated using eq. (5) by an iterative method involving the ionic strength, 1, I = 3Tc, + 3TcZo4 - 4 [CaC204]. Activity coefficients of the divalent cation and anion were assumed equal and were obtained from the extended form of the Debye-Htickel equation proposed by Davies24) : -logf,
3. Results and discussion
OXALATE
= AZ*
I*
~ l+P
-0.3 I
.
Results are summarized in table 1 and the mean K value of 1537 1 mole-’ was used in the calculation of ionic species in all growth experiments. The presence of only the 1 :l complex was also established recently by Armitage and DunsmoreZ6) in a study of this system as a function of ionic strength (I = 0.02 to 0.20). The association constant calculated by Money and Davies*‘) at infinite dilution using the data of Scholder* “) was 1000 1 mole- ’ ; however, this value was obtained from conductivity measurements on only a few saturated solutions where equilibrated water was used and thus large solvent and hydrolysis corrections were involved. The value reported by Gordievskii et aLz9), 1180 1 mole-l, determined using an ion-exchange electrode method, is also subject to error since measurements were made in solutions where the product of total calcium and oxalate was appreciably higher than the value required for spontaneous precipitation of calcium oxalate. In the present work no evidence was
270
G. H. NANCOLLAS
AND TABLE
~~_.
Conductivity
-. T (XlfFM)
TV4 (~10~ M)
3.650 5.753 6.210 6.230 6.254 6.706+ 6.734+ 7.195 7.852 8.574 9.100 10.76* 10.88* 15.76 16.13 23.11
7.180 7.137 6.300 5.236 3.899 4.552** 3.186** 7.108 7.852 8.574 7.070 10.76** 10.88** 15.76 16.13 23.1 I
of calcium
CaZ+ (x IO5 M) 3.353 5.296 5.776 5.853 5.961 6.373 6.474 6.641 7.190 7.776 8.419 9.624 9.746 13.64 13.84 19.16
G.
L. GARDNER
I
oxalate
c2042(x IO5 M) 6.884 6.681 5.866 4.859 3.606 4.218 2.926 6.555 7.190 1.776 6.389 9.624 9.146 13.64 13.84 19.16
solutions
at 25 C
x (obs.) x 10s (ohm-’ cm-‘)
x (Cal.) x lo5 (ohm-’ cm-‘)
2.6080 3.0910 3.0120 2.7153 2.4812 3.0201 2.6541 3.425 I 3.7420 4.1461 3.8656 5.6540 5.7200 7.2420 7.3770 10.340
2.6851 3.2090 3.1238 2.8730 2.5571 3.1060 2.7210 3.5676 3.9122 4.3510 4.0405 5.9441 6.0095 7.7771 1.9574 11.330
K (I mole1510 1540 1525 1565 I596 1463 1601 1530 1555 1614 I540 1534 1494 1482 1568 1474
Mean K = 1537 + 45 (standard deviation) * T,, refers to CaCl,; all others are Ca(N03)2. ** T c2o4 refers to KzCz04; all others are NazCz04. TABLE
Crystallization Expt.
No.
2v + 6v f 21MS Iv 5v 4v 1% 19v* 17” l8v 7v 9” 8v 15” 2&w* 12v 13v 2Ov 27,s 3OMS 25Ms 23v 24v 31MS 28~s 29ws
of calcium
oxalate
2
at 25 ‘C; equivalent
T (x IOC5M)
Ca*+ (x IO5 M)
I (x IO4 M)
7.070 7.744 7.771 8.03 I 8.412 8.443 10.15 10.16 10.28 10.33 10.70 10.73 10.81 12.12 15.59 22.73 23.28 24.24 7.718 7.724 7.833 8.000 8.177 15.22 16.05 22.99
6.462 7.030 7.052 7.270 7.660 7.676 9.005 9.013 9.109 9.152 9.448 9.472 9.536 10.57 13.23 18.31 18.61 19.38 7.008 7.013 7.104 7.200 7.391 12.95 13.58 18.57
3.985 4.345 4.360 4.511 4.736 4.761 5.682 5.630 5.683 5.695 5.903 5.919 5.960 6.65 1 8.396 I I .87 12.10 12.60 4.340 4.344 4.403 4.474 4.584 8.21 I 8.635 12.02
T,, = Total initial calcium, Ca Z+ = initial free calcium concentration. * V = Vibromix stirring, MS = magnetic stirring. * Rate of stirring decreased; in case of magnetic stirring from normal
concentrations Seed suspension
A-l A A-3 A-l A-l A-l A-2 A-2 A-2 A-2 A-2 A-2 A-2 A-2 A-3 A-2 A-3 A-2 B-l B-l B-l B-l B-l B-l B-l B-l
of calcium Seed cont. (mg/lOO ml)
1.70 17.4 14.6 0.33 3.40 1.39 1.80 8.70 8.85 17.4 8.22 8.20 8.31 1.87 14.6 8.20 12.4 8.70 44.2 44.2 44.8 91.5 23.4 89.0 46.0 44.0
200 rpm to 75 rpm.
and oxalate
ions
kx10-2 [I mole-’ min- ’ (mg seed/l00 ml)-‘] 6.1 5.6 8.2 4.8 5.3 5.7 5.0 7.3 5.6 6.9 6.2 4.4 6.1 5.2 5.1 5.6 5.4 6.9 0.95 0.93 I.10 1.20 0.85 0.65 0.45 0.40
I)
KINETICS
OF
CRYSTAL
GROWTH
OF
CALCIUM
OXALATE
271
MONOHYDRATE
found for the existence of higher order complexes [e.g., Ca(C,0JZ2 -1 which were recently reported by Hasegawa, Maki and Sekine3 ‘) in a study of calcium oxalate complexing in 1.0 M sodium perchlorate (25 “C) solutions where a liquid-liquid distribution method was used. Oxalate to calcium concentration ratios were close to one in the present study while ratios of lo3 to lo5 were used by Sekine, and this could account for the formation of a I :2 complex. solubility product KsP The thermodynamic (= [Ca2+] [C202-Ifi) of calcium oxalate monohydrate was determined by allowing crystal growth experiments to proceed to equilibrium, and at 25 “C the mean value obtained was 2.00 f 0.1 x 10m9 mole’ le2 which is in good agreement with McComas and Rieman’s value31) of 1.87 x 10m9 mole’ le2. Other literature values on the solubility of calcium oxalate range from 4.46 to 6.70 x lo-’ M and are uncorrected for complex formation. In addition, the wide range of reported values probably reflects the formation of more than one crystalline phase, since it is known that precipitation of calcium oxalate at room temperature yields a mixture of the monohydrate and dihydrate forms with the slow transformation of the latter phase into the stable monohydrate form32). At the working pH of 6.5 the concentration of the species HC,O,was negligible (< 3 x 10e7 M) and for growth experiments, where supersaturated solutions contained equivalent amounts of calcium and oxalate ions ([Ca2+] = [C,O,‘-I), the initial ionic concentrations were given by: Tc,, = Kf: obtained
[Ca2+]2+[Ca2+]
i,
(7)
from eqs. (2) and (8),
Tc,, = qc204), =
CCa2+li+ CCGOJi.
(8)
represent Tcai and Tcc204)i
the stoichiometric initial concentrations of calcium and oxalate respectively. Using an iterative procedure involving the ionic strength, I, the change in equivalent conductance of calcium oxalate was calculated at various times during a growth experiment. The change in concentration of free calcium or oxalate ions was then obtained from: A [Ca2+]
lo3 Ax = 2/1 , C&204
(9)
where AX represents the change in specific conductivity, and the concentrations are expressed in mole I- ‘.
15
10
5 TIME
(min)
Fig. 1. Growth curves for calcium oxalate Expt. 7”; (a) Expt. 8”; (0) Expt. 9”.
monohydrate.
(0)
A summary of the results obtained for crystal growth experiments is given in table 2 and plots of concentration versus time are shown in fig. 1. For the supersaturation range studied, S = ([Ca2+]i-
[Ca2+]o)/[Ca2+]o
= 0.2
t0
4.0,
the growth in 1: 1 solutions was characterized by an initial fast period followed by a fall in conductivity with time during which the rate of crystallization follows an equation of the form: -d[Ca’+] dt
= ks ([Ca”]
- [Ca2+]0)2,
where [Ca’+] is the ionic concentration of calcium ions (= ionic oxalate concentrations) at time t, [Ca2+], is the ionic solubility under the experimental conditions, s is the surface area of the added seed crystals and k is the observed rate constant for crystal growth. A similar relationship has been useful in the analysis of the growth of magnesium oxalate33) and barium oxalate dihydrate34), as well as a number of bivalent metal sulfates3 5P3“). Supersaturated solutions with total calcium concentrations greater than 2.0 x lop4 M were not stable for more than 30 min and the resulting slow change in conductance with time indicated that spontaneous nucleation of calcium oxalate had taken place. The initial surge in the presence of seed crystals, lasting for IO-20 % of the growth period, probably results from secondary nucleation; the surge was found to increase when large amounts of additives
272
G.
H. NANCOLLAS
were present in solution (e.g. > 5.0 x IO- ’ M Na,P,O,). This observation is consistent with the suggestion made by Sears37) that impurities reduce the energy requirements for two-dimensional nucleation resulting in surface nucleation and polynuclear growth. Similar surge phenomena were observed in the crystal growth of a number of metal sulfates35). In contrast, results from this laboratory on the seeded growth of CaS0,.2H,036) and MgC,0,33) have demonstrated the existence of induction periods, preceding normal growth when the amount of seed crystals was reduced or the supersaturation was increased above rather well-defined limiting values. Microscopicevidence indicated that secondary surface nucleation occurred during the induction period. Although such effects have not been observed under the present experimental conditions, in the case of calcium oxalate, it is not possible to rule out the existence of critical limits of seed concentration and supersaturation beyond which induction periods might be observed. The results of calcium oxalate crystallization experiments at various temperatures from 15 to 45 “C are plotted according to the integrated form of eq. (10) in fig. 2. A plot of log k versus l/T was linear with a slope corresponding to an activation energy for growth, E, = 11.7 * 1.O kcal mole-‘; the magnitude of E, and the negligible change in rate constant with mode and rate of stirring (see table 2) rules out bulk diffusion of electrolyte to the crystal surface as the rate controlling step and suggests an interface mechanism for the growth of calcium oxalate monohydrate. On the assumption that the crystal surface is surrounded by a hydrated monolayer, the observed kinetics of crystallization may be explained in terms of the stationary concentrations in the adsorbed [Ca’+] and [C,0,2-) phase, growth occurring through the simultaneous dehydration of pairs of calcium and oxalate ions at the active growth sites38). Another model for crystal growth involves the adsorption of ions on the crystal surface with partial or complete dehydration and subsequent surface diffusion to active growth positions. Kahlweit and Reich16) have shown that the quadratic dependence for growth [eq. (lo)] can be explained in terms of a rate determining step involving the dehydration of ions at kink sites. Walton39) has obtained an equation, essentially the same as eq. (lo), under conditions of equivalent lattice ionic concentrations by
AND
G.
L. GARDNER
TIME
(mid
Fig. 2. Plots of the integrated form of eq. (10) for the growth of calcium oxalate monohydrate at various temperatures. (0) Expt. 43,, 15 “C; (0) Expt. 17,, 25 “C; (a) Expt. 35,, 35 “C; (a) Expt. 40v, 45 “C.
TIME (mid
Fig. 26.~;
3.
Plots of the integrated (CI) Expt. 32Ms.
form
of eq.
(I 1) for: (i:) Expt.
using a Temkin adsorption isotherm to express surface concentrations. Burton, Cabrera and Frank’s dislocation model for crystal growth4’), originally formulated for growth from the vapor phase and later applied to the solution case by Bennema41), is also consistent with an equation of the form given in eq. (10) at low supersaturations. A number of experiments, made with non-equivalent concentrations of calcium and oxalate ions, are summarized in table 3. The rate of crystallization can be written : -dN/dt
= k’sN2,
where N is the number oxalate to be deposited
(11) of moles per liter of calcium from the supersaturated solu-
KINETICS
OF
CRYSTAL
GROWTH
OF
CALCIUM
Expt. No.
of calcium
* Larger
k’ values
( &?M)
10.28 19.22 21.94 23.39 46.86 7.718 22.40 7.413 may
at 25 “C using
TC,
(x105M)
17v 45v 11” IOV 14” 27hls 26~s 32Ms
oxalate
reflect
secundary
nucleation
non-equivalent czo42(~10~ M)
9.109 17.52 19.65 18.51 39.34 7.008 20.76 5.830
9.109 7.818 8.723 20.75 15.95 7.008 5.850 20.56
Expt. No.
of calcium TC,
oxalate
50”
NTM-TP NTA PMAA NaPSS NaDS
7.718 7.767 7.713 7.638 7.519 7.749 7.518 7.678 7.732 7.806 7.883 23.28 23.75 23.61 23.11 24.14 23.75 23.54 24.59 24.29 23.34 23.88 23.44 12.00 9.690 7.170 9.846 9.822 9.934 7.249 9.803
nitrilo (trimethylenephosphonic nitrilotriacetic acid polymethylene acrylic acid sodium polystyrene sulfonate sodium dodecylsufonate
Seed cont. (mg/lOO ml)
concentrations k’x lo-” [1 mole-’ min-’ (mg seed/l00 ml)-‘]
8.85 (A-2) 8.35 (A-2) 8.45 (A-2) 8.40 (A-2) 8.40 (A-2) 44.2 (B-l) 42.8 (B-l) 42.4 (B-l)
5.6 8.7 8.6 6.0 16.0* 0.95 1.36 1.10
4
at 25 “C in the presence
Additive
ADS
(xl0’M)
27Ms 7OMS 69~s 68hls 67~s 66ms 65~s 7lMS 72hls 73,w 74kls 13” 51” 52v 53” 56~ 54” 55” 61, 60~ 57v 62~ 58~ 59v 46~ 63~ 47, 48~ 49v 64~
and oxalate
at the high supersaturation. TABLE
Crystallization
calcium
Ca2+ (~10~ M)
10.28 9.650 11.01 25.63 23.47 7.718 7.485 22.17
273
MONOHYDRATE
3
TABLE
Crystallization
OXALATE
(M) _
_
Na4P207 Na4P& Na4P2G7 Na4P1G7 NaJ’& Na4P207 NTM-TP NTA
2.92 x 4.35 x 7.19 x 1.42x 4.90 x 1.90x 9.50 x 1.08 x 5.97 x 5.88 x _
lo-* lo- * 10-E lo-’ 10-7 10-e lo-’ lO-6 10-7 lo-’
4.77 x 1.18x 5.74x 6.45 x 8.65 x 7.85 x 8.14x 2.02 x 3.88 x 4.96 x
10-e lo-’ 10-5 lO-5 lO-5 lO-5 lO-6 lo- 5 lO-5 lO-6
Na5PJG10 SnF2 _ NaJ’Kb Nad’z07 NaJ’A Na4P4Q 2 KH2P04 K&G, PMAA PMAA PMAA NaPSS Gelatin Gelatin NaDS NaDS NaDS Fluoroscein Fluoroscein Tetrabromofluoroscein Tetrabromofluoroscein
of various
additives
Seed cont. (mg/lOO ml)
(Tca/Tc,o, _
- = 1)
k*x lO-2 @mole-’ min-’ (mg seed/l00 ml)-‘]
60 ppm 91 ppm 7.60x lO-6 2.20 x 10-S 7.55 x lo- 5 5.05x10-5 1.28 x lo-‘+ 7.38 x lO-5
44.2 (B-l) 44.4 (B-l) 44.1 (B-l) 43.6 (B-l) 43.0 (El) 44.3 (B-l) 43.3 (B-l) 44.0 (B-l) 44.0 (B-l) 44.6 (B-l) 45.0 (B-l) 12.4 (A-3) 16.8 (A-3) 16.8 (A-3) 16.4 (A-3) 17.0 (A-3) 16.8 (A-3) 16.7 (A-3) 6.9 (A-3) 17.1 (A-3) 16.5 (A-3) 16.8 (A-3) 12.5 (A-3) 12.5 (A-3) 16.5 (A-3) 18.2 (A-3) 16.9 (A-3) 16.8 (A-3) 17.0 (A-3) 11 .O (A-3)
0.95 0.84 0.50 0.53 0.25 0.14 0.014 0.15 0.85 0.043 0.85 5.4 1.01 0.32 0.11 1.24 4.5 3.5 0.61 0.35 0.41 1.84 3.4 1.7 6.7 5.6 3.0 2.3 2.2 2.1
1.36 x lO-4
16.8 (A-3)
2.4
acid) Na4P20, sodium pyrosphosphate NaSPsO1c sodium tripolyphosphate Na4P40i2 sodium tetrametaphosphate K&O, potassium pyrosulfate
274
G.
H. NANCOLLAS
6.5
TIME
(mid
Fig. 4. Plots of [Ca’+] versus time. Crystallization in the presence of sodium pyrophosphate. (0) Expt. 65; (0) Expt. 66; (h) Expt. 67; (0) Expt. 68; (n) Expt. 70; (W) Expt. 25.
tion before equilibrium is reached. calculated from the relationship
Values
of N were
fi([Ca’+] -N) ([C,Oi-] - N) = K,, = 2.00 x 10 -9 mole’ 1F2 . It can be seen from the plot of the integrated form of eq. (11) in fig. 3 that the equation satisfactorily represents the crystal growth results in non-stoichiometric calcium oxalate supersaturated solutions. The rate constants in table 3 are appreciably greater when the cation is in excess, indicating that the anion is preferentially adsorbed onto the crystal surface in solutions containing equivalent concentrations of lattice ions. Moreover, the existence of a highly charged surface could explain the marked effects observed with additives discussed later. It is interesting to note that in the case of calcium sulfate dihydrate an increase in growth rate was observed with excess sulfate ions, while with both barium sulfate and silver chloride k’ was smaller than k under conditions both of excess lattice cation and anion. The quadratic expression for the rate of growth of calcium oxalate in pure supersaturated solutions may
AND
G.
L.
GARDNER
also be used to describe results of experiments made in the presence of additives or impurities. These results are summarized in table 4. The strong inhibiting effect of sodium pyrophosphate is illustrated in fig. 4. Sodium tripolyphosphate, nitrilotri (methylenephosphoric acid) sodium tetrametaphosphate and various polyelectrolytes were also effective inhibitors of growth while nitrilotriacetic acid, EDTA, potassium dihydrogen phosphate and potassium pyrosulfate showed negligible effects. It can be seen in table 4 that in all cases the concentration of each additive was sufficiently low that the effective reduction in free calcium ion, through complexation with additive, was negligible and could not account for the observed rate reduction. Compounds containing the structural -P-O-Pand units -N-CH,-P-O appear to be strongly adsorbed onto the surface of CaC20,. H,O with the resulting decrease or complete inhibition of growth. Sears3’) has suggested that the impurites are adsorbed onto the growth steps and consequently, poison the active growth sites, or kinks and are thereby incorporated into the crystal lattice as growth proceeds. Cabrera and Vermilyea42) propose that the adsorption of impurities takes place on the plane regions between the steps thus reducing the step velocity, since the growth front is forced to pass over the adsorbed impurities. As a result, the latter will become buried into the crystal surface. The extensive work done by Kolthoff and Sande1143944) on the effects of coprecipitation in the precipitation of calcium oxalate also adds supporting evidence for the incorporation of impurites in the growing crystal. The heat change due to the precipitation of calcium oxalate, Aqp, is given by Aq,, = Aq (observed)
- Aq (dilution),
(12)
in which Aq (obs.) and Aq (dil.) are the measured heat changes for precipitation and for concomitant dilution of the added reagent, respectively. The heat of precipitation at infinite dilution, AH,‘, is related to the observed A HP by4 5, : AH, = AH;-3RT2
[A-$+
i]
lnf,,
wheref,, the activity coefficient, is calculated from final ionic strength of the solution. The value of D, solvent dielectric constant, and its temperature pendence have been given by Akerlof46). A plot
(13) the the deof
KINETICS
OF
CRYSTAL
GROWTH
OF
CALCIUM
results
OXALATE
275
MONOHYDRATE
have been obtained
for the calcium
phospha-
tes4’) and calcium fluoride4’). The entropy of solution of CaC,O, . H20 was obtained from the relation: AS0 = (AH:-AG,)/T
= -21.8
e.u.,
where AG,‘, the free energy of solution, from the expression : AGY = -RTln
7al 0
IO
20 TIME
(mid
30
x 16’
Fig. 5. Plot of ICa2+] versus time. HAP seed crystals in 300 ml of solution) added to supersaturated solution cium oxalate ([CaZ+], = [C20.,-]r = 1.918 x 10m4 M).
(15 mg of cal-
3.5 1 * ‘0 x 3.0= $=F 9
2.0-
‘i’ ^o r
IS-
s ‘; P “s .%
K,,
was calculated
= 11.85 kcal mole-‘.
The high negative entropy change for the solution process reflects the increase in solvent structure in the ionic co-spheres of the released lattice ions. Lonsdale has suggested the possibility of epitaxial growth of a number of important biological materials including calcium oxalate and calcium phosphate. Initial experiments involving the growth of calcium oxalate on seed crystals of HAP (fig. 5) showed no spontaneous growth, but rather an induction period of about three hours followed by a slow growth process. Eq. (10) adequately explained the kinetics after approximately 30% of the reaction had elapsed as shown in fig. 6. Either surface or bulk nucleation of calcium oxalate followed by normal growth could be involved. References
l.O-
1) H. J. Arnott, 0.5 -
OS cQ 0 5
2) 0 TIME
Fig. 6. Plot of integrated ment shown in fig. 5.
form
I5 (mid
20
25
3) 4) 5) 6)
30
x Kf’
of eq. (10) for growth
experi-
AHP versus log_/, extrapolated using the required slope of eq. (13) gave 5.33 + 0.1 kcal mole-’ for the heat of solution (AH,’ = -AH,‘). The thermodynamic solubility products for CaC,O, . H,O at 15, 25, 35, and 45 “C, 1.55, 2.00, 2.43, and 2.85x 10m9 mole’ lW2 respectively yield a value AH, ’ = 5.5 + 0.1 when plotted according to the Van ‘t Hoff isochore. The difference in activation energies for growth (11.7 kcal mole- ‘) and dissolution4’) (5.8 kcal mole-r) gives - 5.9 kcal mole-’ as the molar heat of solution. The discrepancy between this value and those obtained from equilibrium measurements clearly indicates differences between the transition states for growth and dissolution. Similar
7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20)
F. G. E. Pautard and H. S. Steinfink, Nature 208 (1965) 1197. C. Sterling, Acta Cryst. 18 (1965) 917. D. J. Sutor, Biochem. J. 122 (1971) 6P. B. T. Murphy and L. N. Pyrah, Brit. J. Urol.34 (1962) 729. F. Catalina and L. Cifuentes, Science 169 (1970) 183. K. Lonsdale, D. J. Sutor and S. Wooley, Brit. J. Urol. 40 (1968) 33. K. Lonsdale, D. J. Sutor and S. Wooley, Brit. J. Urol. 40 (1968) 402. K. Lonsdale, Nature 217 (1968) 56. K. Lonsdale and D. J. Sutor, Kristallografiya 16 (1971) 1210. S. Bisay and H. Fleisch, Experientia 20 (1964) 276. A. E. Alexandev, E. P. Crematy and J. E. Crawford, Australian J. Chem. 21 (1968) 1067. D. J. Sutor, Brit. J. Urol. 41 (1969) 171. A. E. Nielson, Acta Chem. Stand. 14 (1960) 1654. A. G. Walton, Anal. Chim. Acta 29 (1963) 434. K. H. Lieser, Z. Physik. Chem. NF 62 (1968) 168. M. Kahlweit and R. Reich, Ber. Bunsenges. Physik. Chem. 72 (1968) 70. H. Chi and K. Nagashima, Bull. Chem. Sot. Japan 41 (1968) 2054. C. 0. Hutton and W. H. Taft, Mineral. Mag. 34 (1965) 256. W. H. Barrett and H. Hartley, J. Chem. Sot. (1913) 786. G. Jones and R. C. Joseph. J. Am. Chem. Sot. 50 (1928) 1049.
276
G. H.
NANCOLLAS
21) G. M. Bollinger and G. Jones, J. Am. Chem. Sot. 51 (1929) 2407; 53 (1931) 411, 1207. 22) G. H. Nancollas, M. S. Mohan, Arch. Oral Biol. 15 (1970) 731. 23) A. McAuley and G. H. Nancollas, J. Chem. Sot. (1963) 989. 24) G. H. Nancollas, Interactions in Electrolyte Solutions (Elsevier, Amsterdam, 1966). 25) R. A. Robinson and R. H. Stokes, Electrolyte Solutions (2nd ed., Butterworths, London, 1959). 26) G. M. Armitage and H. S. Dunsmore, J. Inorg. Nucl. Chem. 34 (1972) 2811. 27) R. W. Money and C. W. Davies, 3. Chem. Sot. (1933) 609. 28) R. Scholder, Chem. Ber. 60 (1927) 1510. 29) A. V. Gordievskii, E. L. Filippov, V. S. Shterman and A. S. Krivoshein, Russ. J. Phys. Chem. 42 (1968) 1050. 30) Y. Hasegawa, K. Maki and T. Sekine, Bull. Chem. Sot. Japan 40 (1967) 2811. 31) W. H. McComas and W. Rieman III, J. Am. Chem. Sot. 64 (1942) 2946. 32) E. R. Sandell and I. M. Kolthoff, J. Phys. Chem. 37 (1933) 153. 33) N. Purdie and G. H. Nancollas, Trans. Faraday Sot. 57 (1961) 1. 34) S. T. Liu, Ph. D. thesis, State University of New York at Buffalo, Buffalo, N. Y., 1972.
AND
G.
L.
GARDNER
35) G. H. Nancollas, J. Crystal Growth 3/4 (1968) 335. 36) S. T. Liu and G. H. Nancollas, J. Crystal Growth 6 (1970) 281. 37) G. W. Sears, in: Growth and Perfection of Crystals, Eds. R. H. Doremus, B. W. Roberts and D. Turnbull (Wiley, New York, 1958). 38) C. W. Davies and A. L. Jones, Discussions Faraday Sot. 5 (1949) 103. 39) A. G. Walton, J. Phys. Chem. 67 (1963) 1920. 40) W. K. Burton, N. Cabrera and F. C. Frank, Phil. Trans. Roy. Sot. London A 243 (1951) 299. 41) P. Bennema, J. Crystal Growth 5 (1969) 29. 42) N. Cabrera and D. A. Vermilyea, Growth and Perfection of Crystals (Wiley, New York, 1958). 43) 1. M. Kolthoff and E. B. Sandell, J. Am. Chem. Soc.59 (1937) 1643. 44) 1. M. Kolthoff and E. B. Sandell, J. Phys. Chem. 37 (1933) 443. 45) J. J. Christensen and R. M. lzatt, in: Physical Methods in Aduancedlnorganic Chemistry, Eds. H. A. 0. Hill and P. Day (lnterscience, New York, 1968). 46) G. Akerlof, J. Am. Chem. Sot. 54 (1932) 4130. 47) G. L. Gardner, Ph. D. thesis, State University of New York at Buffalo, Buffalo, N. Y. 1971. 48) R. W. Marshall and G. H. Nancollas, unpublished results.
Growth 21 (1974) 267-216 0 North-Holland
KINETICS OF CRYSTAL
G. H. NANCOLLAS State
University
Publishing
Co.
GROWTH OF CALCIUM
OXALATE
MONOHYDRATE
and G. L. GARDNER
of New York at Buffalo, Chemistry
Department,
Buflalo, New York 14214, U.S.A.
Received 24 September 1973; revised manuscript received 12 November
1973
The growth of calcium oxalate monohydrate crystals in stirred suspensions has been studied by following the changes in ionic conductivity in supersaturated solutions containing both stoichiometric and nonstoichiometric concentrations of lattice ions at various temperatures in the range 1545 “C. The reaction can be represented in all cases by a kinetic equation which involves a quadratic dependence of the rate of crystallization upon the relative supersaturation. This points to a process in which the crystal growth is controlled by a reaction at the crystal surface rather than by diffusion of lattice ions in the bulk solution. The effect of several inorganic phosphates, polyelectrolytes and organic dyes on the growth kinetics has been studied and the heat of solution of calcium oxalate, determined by a direct calorimetric method, 5.33 kcal mole-‘, may be compared with the value, 5.5 kcal mole-‘, calculated from the temperature coefficient of the solubility product. The thermodynamic association constant for the 1 :l calcium oxalate complex at 25 “C, determined by the conductimetric method, is 1537 1 mole-’ and there is no evidence for higher order species under the conditions of the crystallization experiments. The kinetic results are discussed in terms of the important problems relating to the origin and growth of urinary calculi.
1. Introduction The importance of calcium oxalate formation in solution in both analytical and biological fields points to the need for a quantitative study of the ionic interactions in solutions of this electrolyte and also the kinetics of its crystal growth. The crystallographic structurelp3), morphology4Y5), and composition436’7) of urinary stone materials have shown conclusively the presence of calcium oxalate in nearly all kidney and bladder stones. The calcium oxalate is either the only inorganic material involved or forms a major component of the stone together with other phosphate and urate compounds. The occurrence of various crystalline compounds within stone materials indicates the probable importance of epitaxial growth both at the nucleation and later growth stages of the calculi formation. Such epitaxial relationships between calcium oxalate monohydrate (whewellite), calcium oxalate dihydrate (weddellite) and apatite have recently been discussed by Lonsdale**9). Clearly, in addition to the growth kinetics, the inhibition of stone formation is another area of active interest since in normal urine the product of the ionic calcium and oxalate concentrations is more than one hundred times the value found in saturated solutions of the salt. Various studies”-“) have been carried out recently to elucidate this protective mech-
anism by investigating various types of compounds and ions which are effective in preventing or retarding the growth of calculi. Alexandrev, Crematy, and Crawfordl’) in a recent article have studied the effects of a number of polyelectrolytes on the crystallization of calcium oxalate and conclude that certain urinary polymers adsorb on crystal embryoes and on the faces of the growing crystals, thereby reducing the nucleation and the number of growth sites. Previous studies13-’ “) concerned with the growth of calcium oxalate have dealt with spontaneous precipitation which involves both nucleation and crystal growth. Since both processes may be occurring simultaneously, the kinetics of the reactions are difficult to interpret and the results are very irreproducible. In addition, there is an added factor which was not taken into account by these authors since various hydrates such as CaC204 - H,O, CaC,O, * 2Hz0, or a mixture of the two will nucleate out of solution depending upon the initial supersaturation, temperature and calcium to oxalate molar ratio. The kinetics of nucleation and growth of these individual hydrates may be quite different. The present study is concerned with the kinetics of growth of the monohydrate crystals of calcium oxalate from supersaturated solutions. These solutions are inoculated with well-characterized seed crystals and the 267
26%
G.
H.
NANCOLLAS
rate of crystal growth is followed conductimetrically. This method enables the accurate determination of relative changes of ionic concentrations as small as 0.02 %. The effect of changes of temperature, fluid dynamics, calcium to oxalate molar ratio and the presence of surface active materials on the growth rate have also been investigated. In addition, preliminary experiments have been made to study the epitaxial growth of calcium oxalate on seed crystals of hydroxyapatite (hereafter, HAP; Ca,,(PO,),(OH),). 2. Experimental Solutions were prepared by weight using reagent grade chemicals that had been recrystallized once from conductivity water. Calcium analyses were made using the EDTA titration method with murexide as the indicator and potassium oxalate solutions were analyzed by exchanging the alkali metal ion for hydrogen ion on a Dowex 50 ion-exchange resin and titrating the liberated acid with standard base. Seed crystals, “A”, of calcium oxalate monohydrate were prepared by the drop-wise addition of 100 ml of 0.420 M calcium chloride and 100 ml of 0.426 M potassium oxalate to 600 ml of conductivity water maintained at a temperature of 75 “C. The reaction mixture was stirred continuously during the addition which required about 2 hr; the pH was adjusted to approximately 6.20. The seed crystals were digested at 75 “C for 5 hr, washed several times by decantation until free from chloride, and were aged for about 3 months at 25 “C before use. Dilute seed suspensions, A-l, A-2, A-3 and A-4, containing 1.21, 6.0, 11.5 and 0.60 mg of crystals per ml, respectively, were used for growth experiments. Seed crystals “B” were prepared by slowly adding, with stirring, a 0.25 M oxalic acid solution to a suspension of 20 g of calcium carbonate at 85 “C. The acid was added until the pH of the seed suspension remained at 4.00; the crystals were digested for 8 hr and then left overnight in the acid solution. Crystals prepared using the latter procedure were washed with water until the pH of the seed suspension was about 6.50; dilute suspensions B-l and B-2 contained 68.0 and 10.0 mg of crystals per ml, respectively. Crystals from suspensions “A” and “B” viewed under an optical microscope (Unitron Series N-No. 54372), appeared prismatic, often as aggregates, with approximate edge lengths of l-2 urn and 4-5 urn respectively. Replicas viewed under the electron microscope
AND
G.
L.
GARDNER
showed fibrous crystals with some aggregation. X-ray powder diffraction photographs were similar to those reported1 7’18) for CaC,O, . H,O with no evidence for the dihydrate form. Supersaturated solutions for growth experiments were prepared by the slow mixing of solutions of calcium chloride (or calcium nitrate) and sodium oxalate (or potassium oxalate) in a thermostated conductivity cell. The latter, of 300 ml capacity, was of the Hartley and Barrett type”) constructed from Pyrex glass. Carbon dioxide was excluded from the air space above the cell solution by a slow stream of nitrogen presaturated with conductivity water. Stirring was achieved using either a variable speed magnetic stirrer or a vibrating glass disc (Vibromix Model E). Crystal growth was initiated in these stable supersaturated solutions by the rapid addition of aged seed crystals and resistance measurements were made using a Jones and Joseph screened ac bridge*“*i) or a transformer ratio arm bridge (Wayne Kerr Universal Bridge B221). Precautions normally taken for high precision conductivity measurements were followed. In experiments where additives were used, the same procedure was followed and the freshly prepared additive solutions were added to the cell prior to the introduction of the seed crystals. Experiments involving the growth of calcium oxalate on seed crystals of HAP were carried out at 25 “C using solutions of initial supersaturation in the range 2.0 to 3.0 containing equal concentrations of ionic calcium and oxalate ions. After approximately 30 minutes these stable supersaturated solutions of calcium oxalate were inoculated with seed crystals of HAP, prepared and characterized as described previously*‘), and the reaction was monitored following the change in solution resistance over a 24 to 48 hr period. The differential calorimeter used for studying the heat of precipitation of calcium oxalate was similar to those used previously 23,24). The sensors were 100000 ohm thermistors (Veto No. A189, Victory Engineering Corp.) in a dc Wheatstone bridge arrangement. The calibration heaters (19.591 and 19.685 ohm) were wound from enameled No. 30 Evanohm wire in an open helical form. A typical experiment involved placing 4.0 ml of a known concentration of calcium chloride or sodium oxalate in the mixing device of the calorimeter and 304 ml of a solution of either sodium
KINETICS
OF CRYSTAL
GROWTH
OF CALCIUM
oxalate or calcium chloride in the Dewar flask. After equilibration overnight (25 f 0.005 “C), the precipitation reaction was initiated by opening the mixing device and the corresponding temperature change was recorded; the reaction was effectively complete within 1 min. Two or three electrical calibrations were made before and after each experiment, the current being adjusted so that the corresponding recorder deflection for the electrical calibration was approximately the same as that anticipated for the precipitation reaction. A heat of dilution experiment, into water, was performed simultaneously in each case in the second Dewar of the differential calorimeter; temperature changes of the order 2.5 x lo-’ “C were measured with a sensitivity of approximately 1.O x 10e4 “C. The extent of ion association in solutions of calcium oxalate was calculated from the steady resistance readings in the supersaturated and subsaturated solutions of the salt. In addition, further experiments were made in which weight buret additions of either calcium nitrate or potassium oxalate were made into a weighed amount of conductivity water. The other reagent was then added in small amounts and the resistance recorded. All numerical calculations were made using either an IBM 7044 or a CDC 6400 computer.
In order to be able to analyse the results of the crystal growth experiments, it is necessary to take into account the association of calcium and oxalate ions in reaction (1) : Ca& +
C204& * CaC204(,,,.
The thermodynamic K=
association
CCaC2041 [Ca”]
[C,O:-]
(1) constant
is given by:
1 f?
where fi is the activity coefficient of a divalent ion. The measured specific conductivity, x, of solutions containing mixtures of calcium nitrate (Tc,, total molar concentration) and potassium oxalate (Tc20J can be written IO3 x (observed) +2Tc,
&(No3)2
If no ion
= 2TcZo4 AK2cZo4
-Wa~2~41 &aC204,
where K is the specific and A, the equivalent tance.
(3) conduc-
MONOHYDRATE
association
specific conductance lo3 x (calculated)
took
269
place,
the
calculated
would be given by = 2TcZo4 A KS204
+=Ca
fka(No~jr
(4) The concentration [CaC,O,]
of complex
is therefore
= IO3 [x (calculated-x
(5)
Equivalent conductivity values (A) were using appropriate Onsager equations’ “) A Ca(NOs)z = A CaCl* = A NazCz04 = A KzCz0.1 = A CaC204 =
given by:
(observed)]/2
+A CaC20,.
130.95- 143.7 135X5- 145.3 124.25 - 145.0 147.65 - 152.1 133.65-243.1
calculated
I+, I+, I+, I+, If.
Values of [CaC,O,] were calculated using eq. (5) by an iterative method involving the ionic strength, 1, I = 3Tc, + 3TcZo4 - 4 [CaC204]. Activity coefficients of the divalent cation and anion were assumed equal and were obtained from the extended form of the Debye-Htickel equation proposed by Davies24) : -logf,
3. Results and discussion
OXALATE
= AZ*
I*
~ l+P
-0.3 I
.
Results are summarized in table 1 and the mean K value of 1537 1 mole-’ was used in the calculation of ionic species in all growth experiments. The presence of only the 1 :l complex was also established recently by Armitage and DunsmoreZ6) in a study of this system as a function of ionic strength (I = 0.02 to 0.20). The association constant calculated by Money and Davies*‘) at infinite dilution using the data of Scholder* “) was 1000 1 mole- ’ ; however, this value was obtained from conductivity measurements on only a few saturated solutions where equilibrated water was used and thus large solvent and hydrolysis corrections were involved. The value reported by Gordievskii et aLz9), 1180 1 mole-l, determined using an ion-exchange electrode method, is also subject to error since measurements were made in solutions where the product of total calcium and oxalate was appreciably higher than the value required for spontaneous precipitation of calcium oxalate. In the present work no evidence was
270
G. H. NANCOLLAS
AND TABLE
~~_.
Conductivity
-. T (XlfFM)
TV4 (~10~ M)
3.650 5.753 6.210 6.230 6.254 6.706+ 6.734+ 7.195 7.852 8.574 9.100 10.76* 10.88* 15.76 16.13 23.11
7.180 7.137 6.300 5.236 3.899 4.552** 3.186** 7.108 7.852 8.574 7.070 10.76** 10.88** 15.76 16.13 23.1 I
of calcium
CaZ+ (x IO5 M) 3.353 5.296 5.776 5.853 5.961 6.373 6.474 6.641 7.190 7.776 8.419 9.624 9.746 13.64 13.84 19.16
G.
L. GARDNER
I
oxalate
c2042(x IO5 M) 6.884 6.681 5.866 4.859 3.606 4.218 2.926 6.555 7.190 1.776 6.389 9.624 9.146 13.64 13.84 19.16
solutions
at 25 C
x (obs.) x 10s (ohm-’ cm-‘)
x (Cal.) x lo5 (ohm-’ cm-‘)
2.6080 3.0910 3.0120 2.7153 2.4812 3.0201 2.6541 3.425 I 3.7420 4.1461 3.8656 5.6540 5.7200 7.2420 7.3770 10.340
2.6851 3.2090 3.1238 2.8730 2.5571 3.1060 2.7210 3.5676 3.9122 4.3510 4.0405 5.9441 6.0095 7.7771 1.9574 11.330
K (I mole1510 1540 1525 1565 I596 1463 1601 1530 1555 1614 I540 1534 1494 1482 1568 1474
Mean K = 1537 + 45 (standard deviation) * T,, refers to CaCl,; all others are Ca(N03)2. ** T c2o4 refers to KzCz04; all others are NazCz04. TABLE
Crystallization Expt.
No.
2v + 6v f 21MS Iv 5v 4v 1% 19v* 17” l8v 7v 9” 8v 15” 2&w* 12v 13v 2Ov 27,s 3OMS 25Ms 23v 24v 31MS 28~s 29ws
of calcium
oxalate
2
at 25 ‘C; equivalent
T (x IOC5M)
Ca*+ (x IO5 M)
I (x IO4 M)
7.070 7.744 7.771 8.03 I 8.412 8.443 10.15 10.16 10.28 10.33 10.70 10.73 10.81 12.12 15.59 22.73 23.28 24.24 7.718 7.724 7.833 8.000 8.177 15.22 16.05 22.99
6.462 7.030 7.052 7.270 7.660 7.676 9.005 9.013 9.109 9.152 9.448 9.472 9.536 10.57 13.23 18.31 18.61 19.38 7.008 7.013 7.104 7.200 7.391 12.95 13.58 18.57
3.985 4.345 4.360 4.511 4.736 4.761 5.682 5.630 5.683 5.695 5.903 5.919 5.960 6.65 1 8.396 I I .87 12.10 12.60 4.340 4.344 4.403 4.474 4.584 8.21 I 8.635 12.02
T,, = Total initial calcium, Ca Z+ = initial free calcium concentration. * V = Vibromix stirring, MS = magnetic stirring. * Rate of stirring decreased; in case of magnetic stirring from normal
concentrations Seed suspension
A-l A A-3 A-l A-l A-l A-2 A-2 A-2 A-2 A-2 A-2 A-2 A-2 A-3 A-2 A-3 A-2 B-l B-l B-l B-l B-l B-l B-l B-l
of calcium Seed cont. (mg/lOO ml)
1.70 17.4 14.6 0.33 3.40 1.39 1.80 8.70 8.85 17.4 8.22 8.20 8.31 1.87 14.6 8.20 12.4 8.70 44.2 44.2 44.8 91.5 23.4 89.0 46.0 44.0
200 rpm to 75 rpm.
and oxalate
ions
kx10-2 [I mole-’ min- ’ (mg seed/l00 ml)-‘] 6.1 5.6 8.2 4.8 5.3 5.7 5.0 7.3 5.6 6.9 6.2 4.4 6.1 5.2 5.1 5.6 5.4 6.9 0.95 0.93 I.10 1.20 0.85 0.65 0.45 0.40
I)
KINETICS
OF
CRYSTAL
GROWTH
OF
CALCIUM
OXALATE
271
MONOHYDRATE
found for the existence of higher order complexes [e.g., Ca(C,0JZ2 -1 which were recently reported by Hasegawa, Maki and Sekine3 ‘) in a study of calcium oxalate complexing in 1.0 M sodium perchlorate (25 “C) solutions where a liquid-liquid distribution method was used. Oxalate to calcium concentration ratios were close to one in the present study while ratios of lo3 to lo5 were used by Sekine, and this could account for the formation of a I :2 complex. solubility product KsP The thermodynamic (= [Ca2+] [C202-Ifi) of calcium oxalate monohydrate was determined by allowing crystal growth experiments to proceed to equilibrium, and at 25 “C the mean value obtained was 2.00 f 0.1 x 10m9 mole’ le2 which is in good agreement with McComas and Rieman’s value31) of 1.87 x 10m9 mole’ le2. Other literature values on the solubility of calcium oxalate range from 4.46 to 6.70 x lo-’ M and are uncorrected for complex formation. In addition, the wide range of reported values probably reflects the formation of more than one crystalline phase, since it is known that precipitation of calcium oxalate at room temperature yields a mixture of the monohydrate and dihydrate forms with the slow transformation of the latter phase into the stable monohydrate form32). At the working pH of 6.5 the concentration of the species HC,O,was negligible (< 3 x 10e7 M) and for growth experiments, where supersaturated solutions contained equivalent amounts of calcium and oxalate ions ([Ca2+] = [C,O,‘-I), the initial ionic concentrations were given by: Tc,, = Kf: obtained
[Ca2+]2+[Ca2+]
i,
(7)
from eqs. (2) and (8),
Tc,, = qc204), =
CCa2+li+ CCGOJi.
(8)
represent Tcai and Tcc204)i
the stoichiometric initial concentrations of calcium and oxalate respectively. Using an iterative procedure involving the ionic strength, I, the change in equivalent conductance of calcium oxalate was calculated at various times during a growth experiment. The change in concentration of free calcium or oxalate ions was then obtained from: A [Ca2+]
lo3 Ax = 2/1 , C&204
(9)
where AX represents the change in specific conductivity, and the concentrations are expressed in mole I- ‘.
15
10
5 TIME
(min)
Fig. 1. Growth curves for calcium oxalate Expt. 7”; (a) Expt. 8”; (0) Expt. 9”.
monohydrate.
(0)
A summary of the results obtained for crystal growth experiments is given in table 2 and plots of concentration versus time are shown in fig. 1. For the supersaturation range studied, S = ([Ca2+]i-
[Ca2+]o)/[Ca2+]o
= 0.2
t0
4.0,
the growth in 1: 1 solutions was characterized by an initial fast period followed by a fall in conductivity with time during which the rate of crystallization follows an equation of the form: -d[Ca’+] dt
= ks ([Ca”]
- [Ca2+]0)2,
where [Ca’+] is the ionic concentration of calcium ions (= ionic oxalate concentrations) at time t, [Ca2+], is the ionic solubility under the experimental conditions, s is the surface area of the added seed crystals and k is the observed rate constant for crystal growth. A similar relationship has been useful in the analysis of the growth of magnesium oxalate33) and barium oxalate dihydrate34), as well as a number of bivalent metal sulfates3 5P3“). Supersaturated solutions with total calcium concentrations greater than 2.0 x lop4 M were not stable for more than 30 min and the resulting slow change in conductance with time indicated that spontaneous nucleation of calcium oxalate had taken place. The initial surge in the presence of seed crystals, lasting for IO-20 % of the growth period, probably results from secondary nucleation; the surge was found to increase when large amounts of additives
272
G.
H. NANCOLLAS
were present in solution (e.g. > 5.0 x IO- ’ M Na,P,O,). This observation is consistent with the suggestion made by Sears37) that impurities reduce the energy requirements for two-dimensional nucleation resulting in surface nucleation and polynuclear growth. Similar surge phenomena were observed in the crystal growth of a number of metal sulfates35). In contrast, results from this laboratory on the seeded growth of CaS0,.2H,036) and MgC,0,33) have demonstrated the existence of induction periods, preceding normal growth when the amount of seed crystals was reduced or the supersaturation was increased above rather well-defined limiting values. Microscopicevidence indicated that secondary surface nucleation occurred during the induction period. Although such effects have not been observed under the present experimental conditions, in the case of calcium oxalate, it is not possible to rule out the existence of critical limits of seed concentration and supersaturation beyond which induction periods might be observed. The results of calcium oxalate crystallization experiments at various temperatures from 15 to 45 “C are plotted according to the integrated form of eq. (10) in fig. 2. A plot of log k versus l/T was linear with a slope corresponding to an activation energy for growth, E, = 11.7 * 1.O kcal mole-‘; the magnitude of E, and the negligible change in rate constant with mode and rate of stirring (see table 2) rules out bulk diffusion of electrolyte to the crystal surface as the rate controlling step and suggests an interface mechanism for the growth of calcium oxalate monohydrate. On the assumption that the crystal surface is surrounded by a hydrated monolayer, the observed kinetics of crystallization may be explained in terms of the stationary concentrations in the adsorbed [Ca’+] and [C,0,2-) phase, growth occurring through the simultaneous dehydration of pairs of calcium and oxalate ions at the active growth sites38). Another model for crystal growth involves the adsorption of ions on the crystal surface with partial or complete dehydration and subsequent surface diffusion to active growth positions. Kahlweit and Reich16) have shown that the quadratic dependence for growth [eq. (lo)] can be explained in terms of a rate determining step involving the dehydration of ions at kink sites. Walton39) has obtained an equation, essentially the same as eq. (lo), under conditions of equivalent lattice ionic concentrations by
AND
G.
L. GARDNER
TIME
(mid
Fig. 2. Plots of the integrated form of eq. (10) for the growth of calcium oxalate monohydrate at various temperatures. (0) Expt. 43,, 15 “C; (0) Expt. 17,, 25 “C; (a) Expt. 35,, 35 “C; (a) Expt. 40v, 45 “C.
TIME (mid
Fig. 26.~;
3.
Plots of the integrated (CI) Expt. 32Ms.
form
of eq.
(I 1) for: (i:) Expt.
using a Temkin adsorption isotherm to express surface concentrations. Burton, Cabrera and Frank’s dislocation model for crystal growth4’), originally formulated for growth from the vapor phase and later applied to the solution case by Bennema41), is also consistent with an equation of the form given in eq. (10) at low supersaturations. A number of experiments, made with non-equivalent concentrations of calcium and oxalate ions, are summarized in table 3. The rate of crystallization can be written : -dN/dt
= k’sN2,
where N is the number oxalate to be deposited
(11) of moles per liter of calcium from the supersaturated solu-
KINETICS
OF
CRYSTAL
GROWTH
OF
CALCIUM
Expt. No.
of calcium
* Larger
k’ values
( &?M)
10.28 19.22 21.94 23.39 46.86 7.718 22.40 7.413 may
at 25 “C using
TC,
(x105M)
17v 45v 11” IOV 14” 27hls 26~s 32Ms
oxalate
reflect
secundary
nucleation
non-equivalent czo42(~10~ M)
9.109 17.52 19.65 18.51 39.34 7.008 20.76 5.830
9.109 7.818 8.723 20.75 15.95 7.008 5.850 20.56
Expt. No.
of calcium TC,
oxalate
50”
NTM-TP NTA PMAA NaPSS NaDS
7.718 7.767 7.713 7.638 7.519 7.749 7.518 7.678 7.732 7.806 7.883 23.28 23.75 23.61 23.11 24.14 23.75 23.54 24.59 24.29 23.34 23.88 23.44 12.00 9.690 7.170 9.846 9.822 9.934 7.249 9.803
nitrilo (trimethylenephosphonic nitrilotriacetic acid polymethylene acrylic acid sodium polystyrene sulfonate sodium dodecylsufonate
Seed cont. (mg/lOO ml)
concentrations k’x lo-” [1 mole-’ min-’ (mg seed/l00 ml)-‘]
8.85 (A-2) 8.35 (A-2) 8.45 (A-2) 8.40 (A-2) 8.40 (A-2) 44.2 (B-l) 42.8 (B-l) 42.4 (B-l)
5.6 8.7 8.6 6.0 16.0* 0.95 1.36 1.10
4
at 25 “C in the presence
Additive
ADS
(xl0’M)
27Ms 7OMS 69~s 68hls 67~s 66ms 65~s 7lMS 72hls 73,w 74kls 13” 51” 52v 53” 56~ 54” 55” 61, 60~ 57v 62~ 58~ 59v 46~ 63~ 47, 48~ 49v 64~
and oxalate
at the high supersaturation. TABLE
Crystallization
calcium
Ca2+ (~10~ M)
10.28 9.650 11.01 25.63 23.47 7.718 7.485 22.17
273
MONOHYDRATE
3
TABLE
Crystallization
OXALATE
(M) _
_
Na4P207 Na4P& Na4P2G7 Na4P1G7 NaJ’& Na4P207 NTM-TP NTA
2.92 x 4.35 x 7.19 x 1.42x 4.90 x 1.90x 9.50 x 1.08 x 5.97 x 5.88 x _
lo-* lo- * 10-E lo-’ 10-7 10-e lo-’ lO-6 10-7 lo-’
4.77 x 1.18x 5.74x 6.45 x 8.65 x 7.85 x 8.14x 2.02 x 3.88 x 4.96 x
10-e lo-’ 10-5 lO-5 lO-5 lO-5 lO-6 lo- 5 lO-5 lO-6
Na5PJG10 SnF2 _ NaJ’Kb Nad’z07 NaJ’A Na4P4Q 2 KH2P04 K&G, PMAA PMAA PMAA NaPSS Gelatin Gelatin NaDS NaDS NaDS Fluoroscein Fluoroscein Tetrabromofluoroscein Tetrabromofluoroscein
of various
additives
Seed cont. (mg/lOO ml)
(Tca/Tc,o, _
- = 1)
k*x lO-2 @mole-’ min-’ (mg seed/l00 ml)-‘]
60 ppm 91 ppm 7.60x lO-6 2.20 x 10-S 7.55 x lo- 5 5.05x10-5 1.28 x lo-‘+ 7.38 x lO-5
44.2 (B-l) 44.4 (B-l) 44.1 (B-l) 43.6 (B-l) 43.0 (El) 44.3 (B-l) 43.3 (B-l) 44.0 (B-l) 44.0 (B-l) 44.6 (B-l) 45.0 (B-l) 12.4 (A-3) 16.8 (A-3) 16.8 (A-3) 16.4 (A-3) 17.0 (A-3) 16.8 (A-3) 16.7 (A-3) 6.9 (A-3) 17.1 (A-3) 16.5 (A-3) 16.8 (A-3) 12.5 (A-3) 12.5 (A-3) 16.5 (A-3) 18.2 (A-3) 16.9 (A-3) 16.8 (A-3) 17.0 (A-3) 11 .O (A-3)
0.95 0.84 0.50 0.53 0.25 0.14 0.014 0.15 0.85 0.043 0.85 5.4 1.01 0.32 0.11 1.24 4.5 3.5 0.61 0.35 0.41 1.84 3.4 1.7 6.7 5.6 3.0 2.3 2.2 2.1
1.36 x lO-4
16.8 (A-3)
2.4
acid) Na4P20, sodium pyrosphosphate NaSPsO1c sodium tripolyphosphate Na4P40i2 sodium tetrametaphosphate K&O, potassium pyrosulfate
274
G.
H. NANCOLLAS
6.5
TIME
(mid
Fig. 4. Plots of [Ca’+] versus time. Crystallization in the presence of sodium pyrophosphate. (0) Expt. 65; (0) Expt. 66; (h) Expt. 67; (0) Expt. 68; (n) Expt. 70; (W) Expt. 25.
tion before equilibrium is reached. calculated from the relationship
Values
of N were
fi([Ca’+] -N) ([C,Oi-] - N) = K,, = 2.00 x 10 -9 mole’ 1F2 . It can be seen from the plot of the integrated form of eq. (11) in fig. 3 that the equation satisfactorily represents the crystal growth results in non-stoichiometric calcium oxalate supersaturated solutions. The rate constants in table 3 are appreciably greater when the cation is in excess, indicating that the anion is preferentially adsorbed onto the crystal surface in solutions containing equivalent concentrations of lattice ions. Moreover, the existence of a highly charged surface could explain the marked effects observed with additives discussed later. It is interesting to note that in the case of calcium sulfate dihydrate an increase in growth rate was observed with excess sulfate ions, while with both barium sulfate and silver chloride k’ was smaller than k under conditions both of excess lattice cation and anion. The quadratic expression for the rate of growth of calcium oxalate in pure supersaturated solutions may
AND
G.
L.
GARDNER
also be used to describe results of experiments made in the presence of additives or impurities. These results are summarized in table 4. The strong inhibiting effect of sodium pyrophosphate is illustrated in fig. 4. Sodium tripolyphosphate, nitrilotri (methylenephosphoric acid) sodium tetrametaphosphate and various polyelectrolytes were also effective inhibitors of growth while nitrilotriacetic acid, EDTA, potassium dihydrogen phosphate and potassium pyrosulfate showed negligible effects. It can be seen in table 4 that in all cases the concentration of each additive was sufficiently low that the effective reduction in free calcium ion, through complexation with additive, was negligible and could not account for the observed rate reduction. Compounds containing the structural -P-O-Pand units -N-CH,-P-O appear to be strongly adsorbed onto the surface of CaC20,. H,O with the resulting decrease or complete inhibition of growth. Sears3’) has suggested that the impurites are adsorbed onto the growth steps and consequently, poison the active growth sites, or kinks and are thereby incorporated into the crystal lattice as growth proceeds. Cabrera and Vermilyea42) propose that the adsorption of impurities takes place on the plane regions between the steps thus reducing the step velocity, since the growth front is forced to pass over the adsorbed impurities. As a result, the latter will become buried into the crystal surface. The extensive work done by Kolthoff and Sande1143944) on the effects of coprecipitation in the precipitation of calcium oxalate also adds supporting evidence for the incorporation of impurites in the growing crystal. The heat change due to the precipitation of calcium oxalate, Aqp, is given by Aq,, = Aq (observed)
- Aq (dilution),
(12)
in which Aq (obs.) and Aq (dil.) are the measured heat changes for precipitation and for concomitant dilution of the added reagent, respectively. The heat of precipitation at infinite dilution, AH,‘, is related to the observed A HP by4 5, : AH, = AH;-3RT2
[A-$+
i]
lnf,,
wheref,, the activity coefficient, is calculated from final ionic strength of the solution. The value of D, solvent dielectric constant, and its temperature pendence have been given by Akerlof46). A plot
(13) the the deof
KINETICS
OF
CRYSTAL
GROWTH
OF
CALCIUM
results
OXALATE
275
MONOHYDRATE
have been obtained
for the calcium
phospha-
tes4’) and calcium fluoride4’). The entropy of solution of CaC,O, . H20 was obtained from the relation: AS0 = (AH:-AG,)/T
= -21.8
e.u.,
where AG,‘, the free energy of solution, from the expression : AGY = -RTln
7al 0
IO
20 TIME
(mid
30
x 16’
Fig. 5. Plot of ICa2+] versus time. HAP seed crystals in 300 ml of solution) added to supersaturated solution cium oxalate ([CaZ+], = [C20.,-]r = 1.918 x 10m4 M).
(15 mg of cal-
3.5 1 * ‘0 x 3.0= $=F 9
2.0-
‘i’ ^o r
IS-
s ‘; P “s .%
K,,
was calculated
= 11.85 kcal mole-‘.
The high negative entropy change for the solution process reflects the increase in solvent structure in the ionic co-spheres of the released lattice ions. Lonsdale has suggested the possibility of epitaxial growth of a number of important biological materials including calcium oxalate and calcium phosphate. Initial experiments involving the growth of calcium oxalate on seed crystals of HAP (fig. 5) showed no spontaneous growth, but rather an induction period of about three hours followed by a slow growth process. Eq. (10) adequately explained the kinetics after approximately 30% of the reaction had elapsed as shown in fig. 6. Either surface or bulk nucleation of calcium oxalate followed by normal growth could be involved. References
l.O-
1) H. J. Arnott, 0.5 -
OS cQ 0 5
2) 0 TIME
Fig. 6. Plot of integrated ment shown in fig. 5.
form
I5 (mid
20
25
3) 4) 5) 6)
30
x Kf’
of eq. (10) for growth
experi-
AHP versus log_/, extrapolated using the required slope of eq. (13) gave 5.33 + 0.1 kcal mole-’ for the heat of solution (AH,’ = -AH,‘). The thermodynamic solubility products for CaC,O, . H,O at 15, 25, 35, and 45 “C, 1.55, 2.00, 2.43, and 2.85x 10m9 mole’ lW2 respectively yield a value AH, ’ = 5.5 + 0.1 when plotted according to the Van ‘t Hoff isochore. The difference in activation energies for growth (11.7 kcal mole- ‘) and dissolution4’) (5.8 kcal mole-r) gives - 5.9 kcal mole-’ as the molar heat of solution. The discrepancy between this value and those obtained from equilibrium measurements clearly indicates differences between the transition states for growth and dissolution. Similar
7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20)
F. G. E. Pautard and H. S. Steinfink, Nature 208 (1965) 1197. C. Sterling, Acta Cryst. 18 (1965) 917. D. J. Sutor, Biochem. J. 122 (1971) 6P. B. T. Murphy and L. N. Pyrah, Brit. J. Urol.34 (1962) 729. F. Catalina and L. Cifuentes, Science 169 (1970) 183. K. Lonsdale, D. J. Sutor and S. Wooley, Brit. J. Urol. 40 (1968) 33. K. Lonsdale, D. J. Sutor and S. Wooley, Brit. J. Urol. 40 (1968) 402. K. Lonsdale, Nature 217 (1968) 56. K. Lonsdale and D. J. Sutor, Kristallografiya 16 (1971) 1210. S. Bisay and H. Fleisch, Experientia 20 (1964) 276. A. E. Alexandev, E. P. Crematy and J. E. Crawford, Australian J. Chem. 21 (1968) 1067. D. J. Sutor, Brit. J. Urol. 41 (1969) 171. A. E. Nielson, Acta Chem. Stand. 14 (1960) 1654. A. G. Walton, Anal. Chim. Acta 29 (1963) 434. K. H. Lieser, Z. Physik. Chem. NF 62 (1968) 168. M. Kahlweit and R. Reich, Ber. Bunsenges. Physik. Chem. 72 (1968) 70. H. Chi and K. Nagashima, Bull. Chem. Sot. Japan 41 (1968) 2054. C. 0. Hutton and W. H. Taft, Mineral. Mag. 34 (1965) 256. W. H. Barrett and H. Hartley, J. Chem. Sot. (1913) 786. G. Jones and R. C. Joseph. J. Am. Chem. Sot. 50 (1928) 1049.
276
G. H.
NANCOLLAS
21) G. M. Bollinger and G. Jones, J. Am. Chem. Sot. 51 (1929) 2407; 53 (1931) 411, 1207. 22) G. H. Nancollas, M. S. Mohan, Arch. Oral Biol. 15 (1970) 731. 23) A. McAuley and G. H. Nancollas, J. Chem. Sot. (1963) 989. 24) G. H. Nancollas, Interactions in Electrolyte Solutions (Elsevier, Amsterdam, 1966). 25) R. A. Robinson and R. H. Stokes, Electrolyte Solutions (2nd ed., Butterworths, London, 1959). 26) G. M. Armitage and H. S. Dunsmore, J. Inorg. Nucl. Chem. 34 (1972) 2811. 27) R. W. Money and C. W. Davies, 3. Chem. Sot. (1933) 609. 28) R. Scholder, Chem. Ber. 60 (1927) 1510. 29) A. V. Gordievskii, E. L. Filippov, V. S. Shterman and A. S. Krivoshein, Russ. J. Phys. Chem. 42 (1968) 1050. 30) Y. Hasegawa, K. Maki and T. Sekine, Bull. Chem. Sot. Japan 40 (1967) 2811. 31) W. H. McComas and W. Rieman III, J. Am. Chem. Sot. 64 (1942) 2946. 32) E. R. Sandell and I. M. Kolthoff, J. Phys. Chem. 37 (1933) 153. 33) N. Purdie and G. H. Nancollas, Trans. Faraday Sot. 57 (1961) 1. 34) S. T. Liu, Ph. D. thesis, State University of New York at Buffalo, Buffalo, N. Y., 1972.
AND
G.
L.
GARDNER
35) G. H. Nancollas, J. Crystal Growth 3/4 (1968) 335. 36) S. T. Liu and G. H. Nancollas, J. Crystal Growth 6 (1970) 281. 37) G. W. Sears, in: Growth and Perfection of Crystals, Eds. R. H. Doremus, B. W. Roberts and D. Turnbull (Wiley, New York, 1958). 38) C. W. Davies and A. L. Jones, Discussions Faraday Sot. 5 (1949) 103. 39) A. G. Walton, J. Phys. Chem. 67 (1963) 1920. 40) W. K. Burton, N. Cabrera and F. C. Frank, Phil. Trans. Roy. Sot. London A 243 (1951) 299. 41) P. Bennema, J. Crystal Growth 5 (1969) 29. 42) N. Cabrera and D. A. Vermilyea, Growth and Perfection of Crystals (Wiley, New York, 1958). 43) 1. M. Kolthoff and E. B. Sandell, J. Am. Chem. Soc.59 (1937) 1643. 44) 1. M. Kolthoff and E. B. Sandell, J. Phys. Chem. 37 (1933) 443. 45) J. J. Christensen and R. M. lzatt, in: Physical Methods in Aduancedlnorganic Chemistry, Eds. H. A. 0. Hill and P. Day (lnterscience, New York, 1968). 46) G. Akerlof, J. Am. Chem. Sot. 54 (1932) 4130. 47) G. L. Gardner, Ph. D. thesis, State University of New York at Buffalo, Buffalo, N. Y. 1971. 48) R. W. Marshall and G. H. Nancollas, unpublished results.