sulfate molar ratio on the growth rate of calcium sulfate dihydrate at constant supersaturation

Journal of Crystal Growth 118 (1992) 287—294 North-Holland

~

CRYSTAL

GROWTH

Influence of calcium/sulfate molar ratio on the growth rate of calcium sulfate dihydrate at constant supersaturation Jingwu Zhang and G.H. Nancollas Department of Chemistry, State University of New York at Buffalo, Buffalo, New York 14214, USA Received 16 September 1991; manuscript received in final form 18 November 1991

The growth kinetics of calcium sulfate dihydrate (CSD) has been investigated using the constant composition method over a range of calcium/sulfate molar ratios, r, in supersaturated solutions (0.17 < r < 5.0, at ionic strength, I = 0.500m with potassium chloride as the supporting electrolyte and 0.08 < r <10.0 at I = 1.00m in sodium chloride medium). In spite of constancy of ion activity product, the rate increases with decreasing Ca/SO 4 molar ratio, indicating that the rate of crystal growth is not merely a function of the thermodynamic driving forces but also depends upon the relative concentrations and characteristics of individual lattice ions.

1. Introduction The growth kinetics of calcium sulfate dihydrate (CSD, CaSO4~2H20) has been investigated by a number of workers at equivalent calcium and sulfate concentration [1—81.However, in most practical applications such as scale formation [9], gypsum wall board manufacture [101 and phosphoric acid production [11], gypsum crystallization proceeds in a non-stoichiometric medium. The nucleation rate of CSD, as demonstrate by Keller et al. [121, markedly depends on the calcium/sulfate molar ratio at a given supersaturation. Since it is the subsequent growth process that transforms these micro-nuclei into macroscopic gypsum crystals, there is considerable interest in investigating the influence of calcium/sulfate ratio on the kinetics of gypsum growth. Although the Pitzer formulations have been successfully applied to evaluate ion activity coefficients in solutions at relatively high ionic strength [13—151,as with other empirical approaches, they are only accurate in systems where the parameters have been determined. In contrast to extensive studies of gypsum solubility in NaC1 media 0022-0248/92/$05.00 © 1992

—

[161,the data in the presence of KC1 over a range of Ca/SO4 ratios, are relatively scattered. It was therefore necessary to determine CSD solubility under conditions similar to those of the kinetics experiments. The growth of CSD probably follows an integration controlled spiral mechanism [6,81, SO it is important to estimate the relative concentrations of lattice ions in the adsorption layer. Since zeta (~)potential measurements may offer a unique assessment of the relative adsorption affinity of lattice ions [171, CSD electrophoretic mobility was determined at different Ca/SO4 ratios.

2. Experimental procedure 2.1. Materials Reagent grade chemicals (Baker) and deionized distilled water were used to prepare stock solutions which were filtered twice through 0.22 ~im Millipore filters. The calcium chloride solution was standardized by EDTA titration and the concentrations of potassium sulfate and sodium sulfate were determine by gravimetric barium sul-

Elsevier Science Publishers B.V. All rights reserved

288

J. Zhang, C,.H. Noncollas

/ influence

of calcium /sulfate molar ratio on growth rate of CSD

~ ______

_____ ______________ ________

ing a laser-Doppler technique (Zetasizer 2C, Malvern) [23]. The CSD crystals were finely ground to achieve a stable suspension and the ~ potentials were calculated using the Smoluchowski equation [171. 2.4. Constant composition studies Two titrant solutions were prepared for each constant composition (CC) experiment [24,25], one containing calcium chloride and potassium chloride at concentrations given by eqs. (1) and (2), respectively,

Fig. I. Scanning electron micrograph of CSD seed crystals.

[CaCl2]t=2[CaC12]+Ce,

(1)

[KCl1t=2[KCl]~2Ce,

(2)

the other containing potassium sulfate, fate precipitation. Potassium chloride and sodium chloride, after overnight drying at 120°C, were used as primary standards. Reagent grade CaSO4 2H2O crystals (Baker) were aged in a saturated solution at 25°C for one month before use as seeds. Crystal morphology is shown in fig. 1; the specific surface area was 1.8 ±0.1 as determined by the BET method (30/70 N2/He, Quantasorb, Quantachrome) [181. 2.2. Solubilily determination The solubility of CSD crystals was determined at Ca/SO4 molar ratios ranging from 0.1 to 10 at an ionic strength, 0.5m, with KCI as supporting electrolyte. Nearly saturated CSD solutions, with compositions estimated using literature solubility products along with Pitzer formulations [19—22], were prepared by mixing CaCl2, K2S04 and KC1 stock solutions and equilibrated with sufficient CSD crystals at 25.0 ±0.1°Cfor one week. After filtration, the solutions were titrated against standard EDTA solution to determine the total calcium concentration using Eriochrome Black T as indicator, 2.3. Electrophoretic mobility measurements Saturated solutions of CSD were used as media for electrophoretic mobility measurements us-

[K~So4],

=

2[K~,so4] + C

.

(3)

-

In the above equations, [CaC12], [KCI], and [K2S04] are the supersaturated solution concentrations, and Ce is the effective CSD titrant concentration. For experiments involving NaCl as the supporting electrolyte, NaCI and Na2SO4 were used in place of KC1 and K2S04. Supersaturated solutions (0.250 L) of CSD were prepared in double jacketted Pyrex vessels thermostatted at 25.0 ±0.1°C. Before use, CSD seed crystals (about 30 mg) were ultrasonicated for 15 s in 1 ml of the supersaturated solution in order to break up aggregates. A calcium dcctrode (Orion, Model 93-20) was used to control titrant addition through a potentiostat (Metrohm 605 pH-meter, 614 Impulsomat, 665 Dosimat, Brinkmann, and BD41 Recorder, Kipp & Zonen) to maintain constant solution composition.

3. Rate determination One of the advantages of the CC method is that a large extent of reaction can be achieved even at low driving forces. As the crystals grow, the effective surface area available for deposition increases, leading to a more rapid titrant addition. Thus, the CC titration curves provide a correlation, which may reflect changes in the

/ Influence of calcium /sulfate molar ratio on

J. Zhang, G.H. Nancollas

crystal habit, between the effective surface area increment and the extent of growth. The total effective surface area of a crystal suspension at time t, S~,may be related to the initial value, S0, by the following empirical equation [261: —

~

growth rate of CSD

289

where M is the molecular mass of gypsum. Substitution of eq. (7) into eq. (5) leads to eq. (8): C dV’dt ~ / (8) =

.

S0 (1

+

MC~V/m~)”

.

(4\

/

,.

—~,0~m1m0,

Integrating eq. (8) with initial condition V= 0 at 0, one obtains eqs. (9) and (10):

t

=

k

,

(9)

J=(Ce/S0)(Vc/t),

where m0 and m are the masses of the crystals initially and at time t, respectively. For isotropic three dimensional growth, p 2/3 [27], while i values of 0 and 1/2 indicate that growth occurs exclusively in one and two directions, respectively [28]. The growth rate, by in eq. terms 2), is given (5): of flux density (mol 5~ m Ce dV/dt S (m/m )P~ (5) =

0

0

where dV/dt is the slope of the titration curve at time t. Normally, p values may be calculated from eq. (5) by rewriting it as eq. (6):

(1

—

MC~V/mo) ~ (1 —p)MC /m0

+

—

—

~ )‘

( )

where Y~is the corrected titrant volume for constant both effective surface From eqs. (9) and (10), J and p canarea. be evaluated simultaneously by using a nonlinear least squares procedure. Moreover, eq. (9) suggests that if the correct p value is used, plots of V~against time t should be linear.

4. Results and discussion Table 1 summarizes the composition of the

JS

dV

0 log

—~—

=

log

-~—

+p log

e

m ,

(6)

0

and plotting log(dV/dt) against log(m/m0). However, this procedure requires the calculation of a number of dV/dt values. As the titration curve is not perfectly smooth, especially when a divalent ion selective electrode is used, significant errors may be involved in the estimation of dV/dt. Thus, at equivalent calcium and sulfate concentrations of 0.043m and Ce 0.02m, a 1% change in calcium activity (corresponding to the electrode threshold valuewould for titrant addition) 250 ml working solution require a 5 ml in titrant aliquot, making it difficult to obtain accurate derivatives. The following approach avoids the use of derivatives thereby yielding more accurate p and J values. The total mass of the crystallites can be calculated from the volume of titrant addition, V, =

m

=

m 0 + MCeV,

(7)

equilibrium solutions, along with K5~values calculated using Pitzer equations [21,22]. K5~plots as a function of Ca/SO4 ratio in fig. 2 show a maximum at r 0.3. These K. values may be s•P expressed as a polynomial function in terms of r: =

.

4

K~2 a0

~ a1(log r)’,

+

=

I

=

(11)

I

where the coefficients, a0 2.662 x i0~, a1 —6.614 x i0~, a2 2.702 x i0~, a3 1.842 x iO~ and a4 —6.216 x i0~, were determined by least squares. 2) is At r5%1, higher the value of K5~ X 10~ about than 2.53 (2.66 x i0~ m2 m determined from the solubility of gypsum in pure water with Pitzer formulations [191. However, the present value is in good agreement with that, 2.63 x i0~ m2, obtained from EMF measurements [29]. A review of solubility product determinations has been recently given by Raju and Atkinson [161. In light of the observed variation =

=

=

=

=

=

of K5~with Ca/SO4 molar ratio, some Pitzer

290

J. Zhang, G.H. Nancollas

Table 1 Solubility and r 0.1014 0.1302 0.1759 0.2640 0.5140 1.000 1.000 1.956 3.886 5.976 10.18

/ Influence of calcium /sulfate molar ratio on growth rate of CSD

f

potentials of gypsum at different Ca/SO 4 molar ratios 2~ SO~ Cl~ Ca 1) (mol kg~1) (mol kg~) (mol kg~’) (mol kg~ 0.01264 0.1247 0.3455 0.1214 0.01389 0.1067 0.3620 0.1764 0.01565 0.08893 0.3763 0.2300 0.01858 0.07036 0.3877 0.2841 0.02500 0.04855 0.3910 0.3439 0.03412 0.03412 0.3773 0.3773 0.03411 0.03411 0.3773 0.3773 0.04755 0.02431 0.3454 0.3919 0.06762 0.01740 0.2906 0.3910 0.08378 0.01402 0.2427 0.3823 0.1126 0.01106 0.1574 0.3604

interaction parameters may need re-evaluation, However, in order to maintain the internal consistency of the widely used data base compiled by Harvie and others [20,21], no correction was made of the Pitzer interaction parameters, and the present K 5~values, expressed by eq. (11), were used

K (10~ 5~ mol2 kg~2)

(mV)

2.672 2.694 2.700 2.703 2.680 2.656 2.655 2.655 2.650 2.584 2.586

3.20 5.35 7.52 8.00 10.5 11.4 11.2 12.6 11.6 12.0

for the calculation of the relative supersaturation in KCI media,

U

=

acaaso a~o K4 2

1/2 —

1.

(12)

sP

Table 2 Summary of experimental conditions for CSD growth Experiment r No. I = 1.00 mol kg_i (NaCI)

Ca 1) (mol kg~

IP a) mol2 kg2) (10~~

~.

j(mol

1 2 3 4 5 6 7 8 9 10 11

0.02058 0.02165 0.02165 0.02334 0.02610 0.03104 0.03621 0.04294 0.04292 0.06093 0.1008

4.41 4.31 4.31 4.24 4.19 4.15 4.15 4.15 4.15 4.20 4.34

0.278 0.263 0.263 0.253 0.241 0.244 0.246 0.248 0.248 0.260 0.287

8.26x 10~ 6.88x106 7.56x 10_6 8.15 x 10_6 5.74x i0~ 5.09x106 6.44x 10~ 3.27x106 4.74x 106 2.92x106 1.18x 1O~

0.02087 0.02106 0.02619 0.03892 0.0768 0.1885

4.65 4.31 4.39 4.11 4.16 4.39

0.356 0.305 0.317 0.274 0.282 0.317

4.34x106 3.22x10~6 1.82x106 1.64x 10—6 1.77<10_6 8.36x107

I

0.1667 0.2 0.2 0.25 0.3333 0.5 0.7 1 1 2 5

=

0.500 mol kg_I (KC1)

12 13 14 15 16 17 a)

s~1m2)

0.075 0.1 0.2 0.5 2 10 IP

=

acaasoa~.jo.

J. Zhang, G.H. Nancollas 2.8

/ Influence of calcium /sulfate molar ratio on

growth rate of CSD

291

-__________________________________________

2.5 0.1

1

10

Fig. 2. Plot of solubility product as a function of Ca/SO

4

molar ratio.

Lattice ion activities were calculated from Pitzer expressions and the water activity was computed from osmotic coefficient datataken [21,22]. In the NaC1 2 was as the value of media, 2.53 X l0~ m in eq. (12) [191. CSD growth kinetics was studied at U 0.3 at Ca/SO4 molar ratios ranging from 0.17 to 5.0 in KC1 supporting electrolyte (I 0.50Dm) and from 0.075 to 10.0 in the NaC1 medium (I im). The experimental conditions are summarized in table 2 and typical CC titration curves are shown in fig. 3. The initial portions of the curves were not employed in the rate calculation, for they are strongly influenced by the preparation of the seed crystals as well as the response time of the sensor electrode. Moreover, at very large extents of =

=

__________________________

_______

60 5Q

40 30

20 10

0

-

10

20 30 Time/mir,

~0

50

Fig. 3. A typical plot of CC titrant addition (•) and the corrected volume (0) against time,

Fig. 4. Scanning electron micrograph of CSD crystals after 14 times of overgrowth (m/m0 = 15), with small needle-like crystals probably due to secondary nucleation.

growth (rn/rn 0> 10), secondary nucleation becomes significant as reflected by the fine crystals revealed by electron microscopy (fig. 4). Thus, the rate calculation was based on data from rn/rn0 2.0 to 9.0. It was found that at p 0.4 ±0.1, a linear correlation was obtained between the corrected volume from eq. (10) and time t, for all Ca/SO4 ratios investigated. A typical corrected titrant curve is shown in fig. 3. This p value suggests that the growth of CSD is preferentially, but not exclusively, two-dimensional. Indeed, scanning electron micrographs show that the crystals became more elongated in the z direction, suggesting a relatively fast growth of the (111) faces as compared with the {120) and {010} faces (figs. 1 and 4). Scanning electron micrographs also revealed that variation of Ca/SO4 ratios did not significantly alter the trend in the crystal habit change in KCI media, in support of the relatively constant p value. In the presence of sodium ions, a similar habit change was observed for r= 1.0—10.0, but the crystals appeared to become more tabular as r decreased from 1.0 to 0.075 (fig. 5). This result is in general agreement with a recent study of Witkamp et al. who obtamed a tabular crystal habit in the presence of sodium ions [8]. =

=

292

J. Zhang, G.H. Nancollas

/ Influence o/cclc!I4m/sulfate molar ratio on

growth rate of CSD

son with K~,is probably due to their relatively stronger adsorption at the gypsum surfaces. The difference in rates in the two media becomes smaller at higher Ca/SO4 ratios as the sodium

%

concentration decreases. It has been well established that the rate of crystal growth controlled by volume diffusion depends on the relative concentrations of lattice ions at a given value of the ionic activity product [31]. However, it has also been found that the growth of well stirred CSD suspensions is controlled by surface processes [1—8].In the present study, the rate of CSD growth at o- 0.25 in the2 stoichiometric is 4 X 10_s mol s~ diffum~ (table 2). If it solutions were controlled by volume sion, the diffusion layer thickness, 6, according to =

Hg. 5. Scanning electron micrograph of CSD ciysiak 0.075 in the NaCI medium (I liii).

~

eq. (14), would be 2.1

x iO~ m:

J=D~C In order to assess the influence of molar ratio on growth rates, the latter were corrected to a relative supersaturation of 0.3 using eq. (13), J~=J(0.3/u)~, (13) based on the previously confirmed parabolic rate law for CSD growth [6,30]. Plots of the corrected rate against r (fig. 6) show marked decreases with increasing r. Moreover, the rate is considerably smaller in NaCI media even though the higher ionic strength may have yielded a higher rate [81. The inhibitory effect of sodium ions, in compari-

12 10

‘38

c

(14)

In this equation, the2 diffusion coefficient, s~ and the solubilityD~,is C~ taken 1 x [30].The i0~ m unreasonably large 6 value 34 molasm’3 suggests that volume diffusion is not rate determining. For a symmetric AB crystal with NaC1 lattice, Zhang and Nancollas have shown that even for surface controlled processes, the growth rate is also a function of the solution stoichiometry [32]. On such a crystal surface, there are two types of kinks on the steps, those ending with an A or B =

ion. The former can only accommodate a B ion and the latter an A ion. After receiving an A ion, the B kink becomes an A kink and vice versa. The step advancement is realized by propagation of the kink sites which requires alternate cation and anion integration. For spiral growth controlled by integration, as is probably the case for gypsum growth [6],a maximum rate of step movement can be achieved only when the overall inte-

14

‘C

5U/6.

6

C)

0 0 0.1

1

10

gration rates for both lattice ions are the same. Otherwise, the kink propagation rate will be determined by the integration of the lattice ions with the lower rate. Since the integration rates of lattice ions are proportional to the integration

Fig. 6. Dependence of rate on the Ca/SO 4 ratio at ~ = 0.3. Ionic strength is 0.5m for KCI (•) and lm for NaCI (0).

frequency and their concentrations in the adsorption layer, a maximum growth rate is expected

.J. Zhang, G.H. Nancollas

/ Influence of calcium /sulfate molar ratio on

io

are the corresponding activation energies [30]. Ed~arises primarily from the need to overcome attractive forces between the adsorbed ion and the surface. If only nearest neighbor interactions are considered, Ede values may be assumed to be the same for both cation and anion, leading to eq. (17):

0

0.05

o.~o

i.bo

iooo

Kad+/Kad_~ I”ad+/Vad_.

Fig. 7. Plot of the zeta potential as a function of Ca/SO

4

molar ratio,

when the ionic ratio, p, defined by eq. (15) is unity. K ~ ad~r, (15) =

(17)

Adsorption and integration are similar in the following aspects: The former involves dehydration of the adsorption site and the lattice ion, and a subsequent diffusion jump to the surface. During the latter process, ions in the adsorption layer may have to lose additional waters of hydration before entering a dehydrated kink site. Thus it is reasonable to assume that the frequencies of adsorption and integration are proportional (eq.

i”_Kad_

where ~ and r’_ are integration frequencies of the cations and anions, respectively, and Kad+ and Kad~ are the corresponding adsorption constants. In the present investigation, a maximum rate was not attained probably because it lay at an even lower r value. In order to verify this hypothesis, electrophoretic mobility measurements were made to estimate the parameters in eq. (15). A plot of zeta potential against r is given in fig. 7. The point of the zero charge (PZC) occurs at Ca/SO4 <0.1, in general agreement with data for BaSO4 for which pBa 6.7, corresponding to [Ba]/[SO4] 0.02 [33]. The search for the exact location of the PZC in the present system is hindered by the possible precipitation of syngenite at even lower r values. Since at the PZC, there ought to be equivalent amounts of calcium and sulfate adsorption, =

=

or

(Kd+/K.d~)rPZC=l a a

Kd*/K.d_>lO. a a

This value may subsequently be used to estimate the v÷/v_ ratio with additional assumptions. The adsorption constant Kad can be expressed by eq. (16), =

293

where ~ad and i~ are the adsorption and desorption frequencies, respectively, and Ead and Ede

15

Kad

growth rate of CSD

Vad/Vde

=

exp [ (Ead —

—

Ede

)/kT 1’

(16)

(18). ji~/~i~

i/ad+/l)ad_,

(18)

Since Kad*/Kad_> 10, eqs. (17) and (18) suggest that v±/ii_> 10. Substitution of Kad*/Kad~ and v~/v_ values into eq. (15) leads to a maximum rate at r < 0.01, well beyond the range of the present study. Generally, dehydration of cations has been considered to be the rate determining step for integration and adsorption [30]. However, a!though less hydrated, the bulky SO~ ions may have to overcome considerable rotational energy barriers to integrate into the crystal lattice. Moreover, the presence of hydration water in CSD may facilitate the integration of calcium ions. Thus, both factors favor smaller integration and adsorption frequencies for the sulfate ions. The influence of solution stoichiometry on the rates of crystal growth and dissolution has already received some attention. Abdul-Rahman and Nancollas found that the growth rates of MgF2 and SrF2 crystals, controlled by surface processes, have maximum values when the lattice ion concentrations match the crystal stoichiometry [34]. Stubièar et al. demonstrated that the growth kinetics of PbF2 was dependent on the Pb/F ratio

294

J. Zhang, G.H. Nancollas

/ Influence of calcium /sulfate molar

in the supersaturated solution [35]. Christoffersen and Christoffersen also showed that the dissolution rate of calcium hydroxyapatite is not governed solely by the thermodynamic driving force [36]. The pH-dependence of the dissolution rates of calcium phosphate phases also reflects the importance of solution composition [37]. These previous findings and the present study clearly demonstrate that the rate of crystal growth is a function not only of the thermodynamic driving force, but also the relative lattice ion concentrations.

Acknowledgements We thank Dr. Sudarsan Mukhopadhyay and Dr. Bing Nan Sun for helpful discussions and Dr. Neil Plummer for providing the program PCPITZ for the speciation calculations. References [1] ST. Liu and G.H. Nancollas, J. Crystal Growth 6 (1969) 281. [2] B.R. Smith and F. Sweett, J. Colloid Interface Sci. 37 (1971) 612. [3] G.H. Nancollas, MM. Reddy and F. Tsai, J. Crystal Growth 20 (1973) 211. [4] ST. Liu and G.H. Nancollas, Talanta 20 (1973) 211. [5] G.M. van Rosmalen, P.J. Daudey and W.G.J. Marchée, J. Crystal Growth 52 (1981) 801. [6] MR. Christoffersen, J. Christoffersen, M.P.C. Weijnen and G. van Rosmalen, J. Crystal Growth 58 (1982) 585. [7] L. Amathieu and R. Boistelle, J. Crystal Growth 88 (1988) 183. [8] G.J. Witkamp, J.P. van der Eerden and G.M. van Rosmalen, J. Crystal Growth 102 (1990) 281. [9] J.C. Cowan and D.J. Weintritt, Water-Formed Scale Dcposits (Gulf PubI, Comp., Houston, TX, 1976). [10] Mi. Ridge, Rev. Pure Appl. Chem. 10 (1960) 243. [11] H.M. Stevens, in: Phosphorus and Its Compounds, Vol. 2, Ed. JR. van Wazer (Interscience, New York, 1961) p. 1025. [12] D.M. Keller, RE. Massey and O.E. Hileman, Jr., Can. J. Chem. 58 (180) 2127.

ratio on growth rate of CSD

[131KS.

Pitzer, J. Phys. Chem. 77 (1973) 268. [14] KS. Pitzer and G. Mayorga, J. Phys. Chem. 77 (1973) [15] KS. Pitzer Pure AppI. Chem. 58 (1986) t599. [16] K.U.G. Raju and G. Atkinson, J. Chem. Eng. Data 35 (1990) 361. [17] R.J. Hunter, Zeta Potential in Colloid Science (Academic Press, London, 1981). [18] P.C. Hiemenz, Principle of Colloid and Surface Chemistry (Dekker New York, 1977) ch. 8. [19] C.H. Culberson, G. Latham and R.G. Bates, J. Phys. Chem. 82 (1973) 2693. [20] CE. Harvie and J.H. Weare, Geochim. Cosmochim. Acta 44 (1980) 981. [2t] CE. Harvie, N. Moller and J.H. Weare, Geochim. Cosmochim. Acta 48 (1984) 723. [22] L.N. Plummer, DL. Parkhurst, G.W. Fleming and S.A. Dunkle, A Computer Program Incorporating Pitzer’s Equations for Calculation of Geochemical Reactions in Brines, US Geological Survey, Water-Resources Investigations Report 88-4153, Reston, Virginia, 1988. [23] BR. Ware, Advan. Colloid Interface Sci. 4 (1974) 1. [24] MB. Tomson and G.H. Nancollas, Science 200 (1978) 1059. [25] J. Zhang and G.H. Nancollas, in: Mineral—Water Interface Geochemistry, Eds. M.F. Hochella, Jr. and A.F. White (Mineralogical Society of America, 1990). [26] J. Christoffersen and M.R. Christoffersen, J. Crystal Growth 35 (1976) 79. [27] G.W. van Oosterhout and G.M. van Rosmalen, J. Crystal Growth 48 (1980) 464. [28]J.P. Barone, G.H. Nancollas and Y. Yoshikawa, J. Crystal Growth 63 (1983) 91. [29] T.H. Lilley and C.C. Briggs, Proc. Roy. Soc. (London) A 349 (1976) 355. [30] A.E. Nielsen, J. Crystal Growth 67 (1984) 289. [31] A.E. Nielsen, Croatica Chem. Acta 53 (1980) 255. [32] J. Zhang, PhD Dissertation, State University of New York at Buffalo (1990) ch. 3; J. Zhang and G.H. Nancollas, to be published. [33] AS. Buchanan and E. Heymann, Proc. Roy. Soc. (London) A 195 (1948) 150. [34] A. Abdul-Rahman, PhD Dissertation, State University of New York at Buffalo (1988); A. Abdul-Rahman and G.H. Nancollas, to be published. [35] N. Stubièar, M. Sèrbak and M. Stubi~ar,J. Crystal Growth 100 (1990) 261. [36] J. Christoffersen and M.R. Christoffersen, J. Crystal Growth 47 (1979) 671. [37] J. Christoffersen and MR. Christoffersen, J. Crystal Growth 57 (1982) 21.