Dissolution kinetics and etch pit studies of potassium aluminium sulphate

Journal of Crystal Growth 61(1983)181—193 North-Holland Publishing Company

181

DISSOLUTION KINETICS AND ETCH PIT STUDIES OF POTASSIUM ALUMINIUM SULPHATE B. VAN DER HOEK, W.J.P. VAN ENCKEVORT and W.H. VAN DER LINDEN RIM Laboratory of Solid State Chemistry, Catholic University, Toernooiveld, 6525 ED Nymegen, The Netherlands

Received 14 June 1982; manuscript received in final form 9 December 1982

The dissolution process of the (111) faces of potash alum is studied, both by microtopographic examinations of the etch pit patterns and by measurement of the dissolution kinetics in a rotating disc crystallizer. Both methods showed that the Cabrera—Levine dissolution theory holds for the two most common dislocation types ending on the (Ill) faces of potash alum. On the basis of the rotating disc experiments, the interfacial supersaturation of the etch pit experiments was roughly estimated. Using this, it was found that at interfacial supersaturations below —0.6% (dislocations with <110) Burgers vector) or below —0.85% (dislocations with (100) Burgers vector) numerous etch pits related to those dislocation types appeared. Below those undersaturations the dissolution process is mainly determined by volume diffusion. From the critical undersaturation, determined in the rotating disc crystallizer, the value of the 2. edge free energy of a step was found to be approximately 0.01 J/m

1. Introduction It is widely accepted that dissolution or evaporation of crystal faces preferentially takes place at surface areas where a dislocation line emerges. This is due to the stress energy of partides which are situated in these strained areas. In the case of a screw dislocation a spiral is formed, while at edge dislocations dissolution proceeds via repeated nucleation. Both mechanisms lead to the formation of an etch pit with the dislocation line as axis. By applying a simple thermodynamical model it was shown [1] that the activation energy for the formation of a negative nucleus around a dislocation completely vanishes below a certain undersaturation, which is called the critical undersaturation z~”(undersaturation L~t< 0). For a dislocation ending more or less perpendicular on the growth face this critical undersaturation is a function of, among others, the length of the Burgers vector b, the shear modulus p. and the edge free energy y. When the activation energy is zero, the formation frequency of negative nuclei around a dislocation outcrop is infinite and volume diffusion determines the dissolution rate; usually a 0022-0248/83/0000—0000/$03.00

©

steep etch pit visible by optical microscopy develops. It was found later [2—4]that this activation energy decreased to a large extent upon surpassing the critical undersaturation, but did not vanish totally. It also turned out [5] that the same critical behaviour occurs for a BCF [6] spiral dissolution mechanism around a screw dislocation. The slope of a spiral etch pit increases slowly between undersaturations of zero and ~ whereas below the critical undersaturation the slope, and thus the etching rate, suddenly increases drastically. It is the aim of this paper to show that the Cabrera and Levine (CL) dissolution theory [1] applies to the dissolution of the (ill) faces of potassium aluminium sulphate crystals. It will be shown that below a certain undersaturation very steep etch pits occur and that the etch rate increases enormously. The critical undersaturation will be determined and from that value the edge free energy y for the <111) faces of alum will be evaluated. In order to achieve this, two kinds of experiment have been performed: (i) Observation of etch pits formed at various undersaturations and measurement of their slopes, (ii) measurement of the dissolution rate as function of the interfacial undersaturation.

1983 North-Holland

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Dissolution kinetics and etch pit studies ofpotash alum

2. Theory

The value of the critical undersaturation follows from the thermodynamical model in refs. [1,4]:

2.]. Dissolution around a dislocation

In an isotropic medium, the stress density u(r) around a dislocation with Burgers vector b and line direction I follows from Hooke’s elasticity law: (1) where p. is the shear modulus and r the distance to the dislocation line. K(4) is given by the expression:

(3) k~iL*I 2~r2~2y2/p.b2K(~), where Q is the molecular volume of a growth particle. Since the value of np.” originates from the thermodynamical model of CL it is independent of the actual dissolution mechanism.

u(r)=p.b2K(~)/8~r2r2,

K(~) (sin2~)/(l =

—

v) + cos2q~,

(2)

where v is the Poisson ratio and 4 the angle between b and I. However, eq. (1) is not applicable close to the dislocation line since the strain density would exceed the elastic limit if b is constant. Therefore, the (upper) value of u in the dislocation core is limited to the so called core energy density u(0) [4,7]. A smooth energy density function is introduced in ref. [4] where u u(0) for small r and u equals eq. (1) for large r. In ref. [5] results are reported of a computer simulation of spiral growth and dissolution around dislocations with such a stress function. The normal case is when stress fields are so strong that b[2 u(0) p. K(4)]°5/4wy> 1.56,

where y is the edge energy of a step. Then, two kinds of dissolution spirals occur around screw dislocations: (i) When undersaturation ~Xp.decreases from 0 to i~p.’ relatively widely spaced spirals (step spacing decreasing from 19 to 4 times the radius of the critical nucleus on an unstressed surface, ,~)occur. (ii) When V~p.I>~ suddenly closely spaced spirals develop; the spacing decreases with increasing b[u(0)]°5/y and can easily be smaller than ,~ at the critical undersaturation. In the case of very weak stress fields, b[ 2 u(0) p. K(~)]° <4~y < 1.56, only the widely spaced spirals, with a spacing between 4r, and 19r,~,can occur. The undersaturation ~p. is defined as the chemical potential difference between the solution and the crystal and thus is negative in the case of dissolution.

2.2. Determination of the interface concentration c,

In the case of dissolution, the undersaturation at the dissolving surface is lower than the bulk undersaturation. This is due to the concentration gradient in the mass transfer boundary layer formed by diffusion of particles from the dissolving face towards the bulk. In order to determine the interface undersaturation the volume resistance 6/D, where 6 is the thickness of the mass transfer boundary layer and D the diffusion coefficient, should be well defined during the experiments. This is achieved by choosing a specific hydrodynamic environment for the dissolving face. In the present case a crystal slice was inbedded in a plastic disc, which was rotated around its perpendicular axis in an undersaturated solution. In the case of a rotating infinite smooth disc in an infinite vessel, the Navier—Stokes equations can be solved [8] and the thickness of the mass transfer boundary layer 6 equals: 6

=

1.61 f(Sc) D1”~3v1”6.i

—

1/2

=

0.5( D/v)”~’36H,

(4) where v is the kinematic viscosity of the solution, w the rotational velocity and 6H the hydrodynamical boundary layer. In our alum solutions the Schmidt number Sc p/D is about 2200 and a correction of f(Sc) 1.02 has to be used in calculating the value of 6 [9]. In his derivation Levich considers a face where the particles are uniformly transferred from the interface into the boundary layer. In the case of dissolution of crystals, the transfer of partides takes place at the steps, but since the step distances are by far smaller than 6 we can assume that eq. (4) is applicable. It is shown in ref. [10] that eq. (4) also holds for finite discs in a finite vessel, provided that the =

=

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specific dimensions (such as the distance of the disc to the wall of the vessel or the radius of the disc) are considerably larger than 6; when the disc surface is rough the height of the protuberances must be less than 6. Eq. (4) holds in the case of laminar flow, i.e. when the Reynolds number Re wr2/~is smaller than l0~—l0~ [8]. The mass flux ~Iimof solute from a dissolving face is due to diffusion as well as due to the occurring diffusion drift flow and implicitly is given by:

The densities Ps 1750 kg/m3 and PL 1072 kg/rn3 are almost constant with respect to temperature. =

=

2.3. Correlation between difference in chemical potential Lip. and supersaturation a

=

—DpLdc/dx+~mc,

~m

(5)

since the net transport of the second component (solvent) is zero. In this equation x is the distance to the face, c the mass fraction of the solute in the solution and PL the density of the solution. The dissolution rate R sli/p5 resulting from the mass flux follows from eq. (5) [11,12]: =

PLD

6 R=—ln

l—cb 1 — c,

(6)

pLD(cb—cI) p~6(l — c~)

Ps

The fundamental driving force for dissolution and etch pit formation is the difference in chemical potential ~p. of the growth units in the solution and in the crystal. By definition zip. is given by: (10)

L~p.=kTln(a/aeq),

where a and aeq are the solute activities in the under(super)saturated solution and the saturated solution, respectively. Since an alum solution consists of an equimolar mixture of Al 2(S04)3 and K2S04 as well as “dissolved” crystal water the ~s of a K2S04 Al2(S04)3 24H~Osolution of molal.

.

itymisequalto: a ~p.=kTln

/

+24kTln~

where p 5 is the density of the crystal and cb and c~ are the mass fractions of the solute in the bulk of the solution (x> 6) and at the crystal face (x 0), respectively. The second approximation holds for small concentration differences. The dissolution rate R can be determined experimentally from rotating disc experiments and the eq. p(4). Provided that the value values ofof6D,follows v, ch,from PL and 5 are known, the value of c, and thus of the interface undersaturation, can be determined for various rotation velocities w and bulk undersaturations. The values of D and r’ for saturated alum solutions are reported in ref. [13]:

a~,eq

mix

(~)saIt.

)mix’

(11)

=

D

(0.0912T+ 2.036) X 10- 10 20
=

(—

[m2/s],

(7)

{m2/s],

(8)

where T is the temperature in °C.The saturation concentration Ceq of alum in water is given in ref. [14]: 4T+ 5.958 x 105T2 ceq 0.0616 + 8.942 x 10 [(kg solute)/(kg solution)] (9) =

.

where aw,mix is the water activity of the solution. Both terms can be evaluated by making use of the first approximation of the RWR equation [15,16]. Using this relation the 4mix following of (whichdependence is directly rethe osmotic coefficient lated to aw,mix [16]) on the osmotic coefficients 1 of the separated solutions of K2S04 and Al2(S04)3 at the ion strength of the mixture was derived: 4~mix= 84~K 2SO4

(6m)

+-gPAl(5o)

(1.2m).

(12)

Using the osmotic coefficients of aluminium sulphate [17] and of potassium sulphate [16], combined with eqs. (11) and (12) and the Gibbs—Duhem equation it turned out that for lower super(under)saturation ~p. is porportional to a: ~p./kT=

3.3[(m

—

—0.1
meq)/(meq)] T

30°C.

=

3.3a, (13)

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3. Surface microtopography of etch pit patterns

Nomarski combined with a mercury light source fitted with a monochromator filter (X 546 nm) has been used. In order to correlate etch pits with dislocations, X-ray diffraction topography after Lang [23] has been employed. Technical details on this method as well as on the preparation of the potash alum slices have been given elsewhere [24,25]. =

3.1. Experimental The dissolution experiments were carried out in a vessel especially designed for easy separation of the crystals from the undersaturated solution, in order to preserve the original surface patterns. The essential point of this vessel, described in ref. [18], is that the solution is covered by a layer of n-hexane of about 5 cm in thickness. During separation of the crystal from the solution at the end of the experiment the crystal passes the n-hexane layer, so that the hexane replaces the adhering solution to a great extent. After removal, the crystal is dipped into n-hexane for a few seconds and subsequently dried by means of a paper tissue. In this way clean surface areas suitable for surface microtopography can be obtained [18—20]. An undersaturation, determined within an adcuracy of 0.05 K, was obtained by saturating the solution followed by an increase of the temperature, which was measured with a thermocouple. The dissolution experiments were carried out as follows: A freshly grown crystal, obtained by standard methods [18,21] is first placed for 2 h in the hexane layer above the solution, in order to get the solution temperature. Then, the alum crystal is lowered into the solution, where it is allowed to dissolve for a period varying from 20 s (Tb ‘~q ~(Tb 4 K) to 30 mm (Tb T~q ~Tb 0.2 K), depending on the undersaturation. The solution in which the crystal rotates is agitated. Further, Tb is the temperature of the solution, ‘~qis the saturation temperature and ~Tb is the bulk supercooling. Finally, the crystal is removed according to the procedure given above. For each experiment a freshly grown crystal had to be used, since slightly dissolved crystals show somewhat curved faces, which implies that the steps generated from the rounded edges of these crystal faces slow down etch pit formation to a great extent. The surface patterns of the slightly dissolved (111) potash alum faces have been observed by means of a Nomarski differential interference contrast microscope [22]. In order to measure the inclinations of the etch pit walls with respect to the (111) surface a two-beam interferometer after —

=

=

—

=

=

3.2. Correlation between etch pits and dislocations In a previous work [21] it was shown by an etch—polish--re-etch method and by a correlation between growth hillocks and etch pits that the well-known triangular pits on <111) potash alum formed after etching in water are strictly related to dislocation outcrops. To establish this once more and to determine the Burgers vectors involved in the etch pit formation the following experiment was carried out: Firstly a good quality alum crystal was etched with water and the position of an isolated etch pit was marked by a droplet of paint (see fig. Ia). An (110) crystal slice, containing the mark, was cut and Lang topographs were made. These X-ray topographs for the (110) (fig. Ib) and the (001) (fig. ic) reflections clearly show that a single dislocation terminates at the position of the mark, i.e. the etch pit. In the neighbouring region, where no other pits were found, no dislocation lines can be ~discerned. The line width of the dislocation related to the etch pit in fig. 1 is typical of most of the dislocation lines observed on X-ray topographs of alum slices using Ag Ka~radiation [25].This means that the Burgers vector of this dislocation line is of unit height, implying that such dislocations are capable of forming etch pits. Other evidence for the fact that unit dislocations induce etch pit formation after etching in water is given by the following observations: (i) The number of etch pits is roughly equal to (and often even higher than) the total number of dislocations revealed by Lang topography on the same face. (ii) The dislocation lines having the lowest possible Burgers vector b <001) [24,251 are marked by pits after etching. (iii) Each growth hillock on (111) alum changes into a pit after dissolution in water [21], while it was dem=

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!~

185

Dissolution kinetics and etch pit studies of potash alum

‘a

______

b

~~tc

Fig. 1. Correlation between etch pits and dislocations: (a) isolated pit formed after dissolution in pure water; (b) Lang topograph of (110) slice (g [110]) containing the pit shown in (a), indicated by arrow; (c) as (b), but g = [001].

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B. van der Hock et a!.

/ Dissolution

onstrated [18] that growth hillocks represent spirals with unit lattice step height and thus can be related to unit Burgers vectors, 3.3. Etch pits patterns in dependence on the bulk undersaturation

In order to study the variation of the surface morphology of as-dissolved (111) alum faces in dependence on the bulk undersaturation, numer-

kinetics and etch pit studies of potash alum

ous freshly grown crystals were etched in well-defined, slightly undersaturated aquous solutions according to the procedure in section 3.1. Some results of these experiments at several undersaturations expressed as supercoolings ziTb are presented in fig. 2. From this figure three undersaturation regions can be distinguished: (i) 0
~___

_

I

___

-~

Fig. 2. Surface morphology of dissohed (lll~potash alum faces in dependence on the hulk undersaturation, indicated in figures.

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187

Dissolution kinetics and etch pit studies of potash alum

z~Tb< 1.8 K. Here a first group of etch pits all

having an equal slope, which increases for increasing ~Tb, is formed (fig. 2c). In the following, these pits will be denoted as generation I pits. (iii) ~Tb> 1.8 K. In this undersaturation domain a second group of pits, denoted as generation II, can

be perceived. Just like generation I, these pits have the same inclination which increases to 7° at infinite .~Tb(figs. 2d—2f). To verify the existence of the three undersaturation regions and to determine the relative amount of pits formed at a given undersaturation with

~~ioC

V

~II ___

~iI

_

~

~

I

— ~ —___

Fig. 3. Comparison of etch pit patterns created after dissolution at the indicated undersaturations (a—c) with the patterns of the same surface area formed after dissolution in water (a’—c’).

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Dissolution kinetics and etch pit studies of potash alum

respect to the total number of dislocation outcrops, the dissolved crystal faces were, after being photographed re-etched in pure water and rephotographed. Some of the results are shown in fig. 3: On crystal surfaces dissolved at low undersaturations, which were almost free of pits, numerous pits are formed after re-etching with water, mdidating the existence of dislocations (figs. 3a and 3a’). For surfaces dissolved at intermediate undersaturations, where only generation I pits were created, additional pits were formed (figs. 3b and 3b’). By these etching studies it was established that generation I pits present about 30% of the total number of dislocation outcrops at the surface, Finally, on crystal surfaces dissolved at higher undersaturations, where both generations I as II were created, in general no additional pits are formed after etching in water (figs. 3c and 3c’). This means that generations I and II together mark almost all dislocation outcrops on the alum (111) faces. By X-ray topography, two kinds of dislocation have been identified unambiguously, namely those with Burgers vectors <110> and <100> [24,25]. The occurrence of these two dislocation types must match the occurrence of generation I and II etch pits. Since b1 > b than around <100> dislocation outcrops (see eq. (1)). This means that for lower undersaturations dissolution preferentially takes place at the stronger stress field, i.e. around <110) dislocations. So, generation I pits are related to <110) dislocations and generation II pits, which are only formed at higher undersaturations, are related to <100> dislocations. The occasionally observed etch pits on low quality crystals at very low undersaturations (~Tb <0.8 K) must be related to higher (non-unit height) Burgers vectors or to a cooperating group of unit dislocations. Finally, it must be mentioned that generation I pits may partly be related to <111>-type disloca tions, although the existence of this dislocation type has never been proven positively [25]. 3.4. Inclination of etch pits in dependence on the bulk undersaturation

For each of the dissolved alum crystals twobeam interferograms, together with interference

contrast micrographs, of representative
—,

________________________________________

.~

_____

~

4

_____

_________________________________________ ______ ___________

_____ ______ ________________

-____ ., —

_____________ ___________________

b

I

______

Fig. 4. Measurement of inclinations of etch pits with respect to the (11 1) plane: (a) Interference contrast micrograph of surface pattern formed after etching at ~T 5 = 2 K. (b) Two-beam interferogram after Nomarski of the same area as (a); note the two different etch pit slopes (types I and II).

B. van der Hock ci at

(cyb~la)—~

~

/ Dissolution

IL

—~

189

kinetics and etch pit studies of potash alum

r1i

~

in water.,,,,~

/

6

7-~it~~

1;

4.

6

0/~

/

(100)

.—-----------

(110)

a. —ø 05%

-

I__ -

--

-

10% ‘——1

I

Fig. 5. Dependence of the slope of pit sides on the bulk undersaturation (scale at top) and the roughly estimated interface undersaturation (scale at bottom) a, for generation I and II pits.

lion I pits is about 0.8 K

(—

2.6%) and for genera-

lion II about 1.8 K (—5.9%). The occurrence of these critical points for etch pit formation gives a strong indication of the applicability of the dissolution model of Cabrera and Levine [1]. For high undersaturations (i.e., pure water) the slope goes up to 6.75°for both etch pit types.

4. Dissolution and growth kinetics 4.1. Experimental

For the measurement of dissolution and growth kinetics about the same experimental set-up as described in ref. [12] was used. A scheme of the rotating disc crystallizer is shown in fig. 6. A glass vessel (volume 7.5 litre) is placed in a 70 litre water thermostat in which the temperature can be kept constant within 0.01 K. The volume of the vessel is large enough to prevent changes in satura-

Fig. 6. Scheme of rotating disc crystallizer: (1) 70 litre water thermostat; (2) 7~ litre glass crystallization vessel; (3) baffles; (4) crystal slice embedded in plastic rotating disc; (5) stirring rod; (6) air thermostat; (7) revolution counter. Courtesy R. Janssen-van Rosmalen.

tion due to growth or dissolution of the crystal. In this vessel a cylindrical plastic matrix the “disc in which a (111) crystal slice is mounted, rotates in the solution. In order to prevent rotation of the solution, baffles are placed in the vessel. Prior to an experiment the disc is allowed to attain the same temperature as the solution by acclimatization in the air thermostat above the vessel. The stirring rod is a polyacetate-covered steel bar and rotation velocities of up to 800 rpm can be reached without oscillation of the disc. The disc was made by pouring epoxy around a <111) slice cut from a single crystal of potash alum into a cylindrically shaped mould [11,12].The disc radius was 13 mm. The linear dissolution rate R could be determined directly by measuring the vertical displacement of the crystal surface with a dial gauge clock (accuracy 0.01 mm). —

—

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Dissolution kinetics and etch pit studies of potash alum

The •under(super)saturation is established by raising (lowering) the temperature in the vessel. 7~qcan be determined The saturation temperature roughly with help of eq. (9); it is determined with an accuracy of 0.01 K by measuring growth and dissolution rates close to the approximated ‘~q’By the same procedure, the value of is checked from time to time. 4.2. Results

Series of mainly dissolution experiments were made by using two solutions with saturation ternperatures (l~q)of 30.48 and 30.53°C, rotation velocities of 25 to 600 rpm and various undercoolings and supercoolings, 3.52> Tb 7•~q> —0.85 K. For the chosen rotation velocities the value of the Reynolds number varied between 400 and 10,000, i.e. within the laminar flow regime. The mass transfer boundary layer thickness ranged from 80 p.m at 25 rpm to 16 p.m at 600 rpm. In fig. 7a the vertical displacement of the crystal surface h is plotted versus the dissolution time t, with h 0 at t 0 for subsequent dissolution experiments with one given crystal. The slope of the line is equal to R and we can conclude that the dissolution rate is constant and not depending on the length of time interval. In fig. 7b a plot of h versus t, obtained from dissolution experiments with two different discs is shown. Here too the —

=

dissolution rate for two different discs is the same. During the dissolution experiments shown in fig. 7a the difference in height of the crystal face with respect to the disc surface ranged from + 0.8 to —0.9 mm. Since the dissolution rate is equivalent for all points of fig. 7a, we conclude that this difference in height has little or no influence on the dissolution rate. So, from fig. 7 it can be concluded that different time intervals or crystal discs do not affect the dissolution rate; this was confirmed by most of our other dissolution rate data. In the case of growth, the scatter in measured growth rates was high, however. Especially the length of the time interval had influence on the growth rate; at smaller time intervals (t <20 h) the growth rate was twice larger than at large time intervals (i> 100 h). Qualitatively, this is in accordance with the results and conclusions reported in

/.

200

=

150

/

/ /

j

~/ ~cb/ ~‘/A ________________

/

/

/

100

__________________

1.5

~m)

£

/

(~Is)

50

/

t(10

~

a)

—~

t(io s) ~

—~

Fig. 7. Perpendicular displacement of crystal face as function of dissolution time interval: (a) ce = 500 rpm, ~T 5 = 0.28 K, consecutive dissolution experiments, using same disc; (b) ~e= 100 rpm, .~T5= 1.5 K, dissolution data of two different discs (•, 0).

25

5.0

if~j (radls)

112 —,

Fig. 8. The ~c dependence of the linear dissolution rate R of the (Ill) faces of alum for various supercoolings (undersaturations) indicated in K.

B. van der Hock ci a!.

/

the dissertation of Human [26], where this scatter was attributed to a decrease in strength of cooperating spiral centres. For every bulk undersaturation ~Tb and to, the displacement of the crystal face was measured in two to five experiments; the dissolution experiments showed little scatter in the dissolution rate, The dissolution rates for various bulk supercoolings .~Thare shown as a function of i/to in fig. 8. In order to verify whether the quasi-linear behaviour of the data shown in fig. 8 points to pure volume diffusion determined dissolution, we will define an overall mass transfer coefficient k101: k101

=

________ —

pL(cb

ceq)

p~R

=

PL(cb

ceq)

—

(14)

In fig. 9 we compare the experimental with the volume mass transfer coefficient k~0,, which follows from eqs. (4) and (6): k~01/V~

—

-

~ 1/6 I.61f(Sc)(l -ceq)’

191

Dissolution kinetics and etch pit studies of potash alum

(15)

When k101 k~01the mass transfer process is totally volume diffusion determined. Although it seems in fig. 8 that R is proportional to %/~and hence that k~01/V~is constant (i.e. volume diffusion determined dissolution), one realizes from fig. 9 that this is not the case, since for low under=

coolings k~0~ deviates considerably from ~ It can be seen that the overall mass transfer coefficient k101 increases rapidly with increasing ~Tb, up to ~Tb I K, where k10~approximates the volume mass transfer ~ Beyond that supercooling the overall mass transfer coefficient increases very slowly. Realizing that the difference between k~0~ and k101 is a measure for the surface mass transfer resistance, it can be seen that at low ~XTb the surface has a large mass transfer resistance, which decreases for increasing ~Tb, while at ~Tb> 1 K the surface has only a small, but significant, mass transfer resistance. In fact, the two regimes of the extended CL dissolution theory can be distinguished:for Above L~p.”(zlTb < 1 K) a large surface resistance dissolution exists, corresponding to shallow etch pits and a low dissolution rate, below ~ (L~T~,> 1 K) dissolution is mainly volume diffusion determined, i.e. steep etch pits and a high dissolution rate. However, at the higher supercoolings still a certain surface mass transfer resistance is present, which indicates that the surface supersaturation is not zero, as would be the case if the process was totally volume diffusion controlled (c ceq). This is in accordance with the extended CL theory [5] because for fast dissolution involving steep etch pits always an interface undersaturation of z~p.” should be present. For the high undercoolings, ~T1, 1.44 to 2.36 K, values of the surface supersaturation a~of about 1% were found from the experimental data in fig. 9 using eq. (6). On the growth side of fig. 9, we see that the overall mass transfer coefficient in the case of =

=

—

growth

~

dIssolutIon

200 rpm (6 25 p.m) is much smaller than the volume mass transfer coefficient. This indicates that the growth process of alum is determined by the surface mass transfer process. to

=

=

2

~ ~,

Finally, in fig. 10 the dissolution and growth rate as a function of the interface supersaturation a (= (c1 ceq)/ceq) is shown. For every point with known R, ~Tb and to the interface concentration can be derived with help of eqs. (4) and (6). In the case of growth, a > 0, we show the —

—

—

-i

~T (K ) 1

2

4

Fig. 9. Overall mass transfer coefficient k1,,1 (solid circles and solid curve) and the volume mass transfer coefficient k~,,1 (dashed curve), both divided by ~ as a function of supercooling ~ For dissolution, k101 is treated as a linear function of i/~ for growth, k,,,1 is shown for a rotation rate of 200 rpm.

growth data with the small measuring time. From fig. 10 we conclude that the dissolution rate increases enormously when the interface undersaturation is high; the two CL regions (shallow

192

B. van der Hock et a!.

diSSolutIon I

I

/

Dissolution kinetics and etch pit studies of potash alum

Using the results of the rotating disc experiments the interface undersaturation a1 was roughly correlated to the bulk undersaturation 0b (~Tb) of Gb. the With soluhelp of this correlation, the tion in which the observed crystals were dissolved

growth .-i__-——j—o--——~

11.

1%

20!.

-100

0/ 0

R(nm/s) -200

0

00 —

Fig. 10. The linear growth and dissolution rates as a function of interface super(under)saturation 0.

was converted into an interface undersaturation a. The measured etch pit inclination is plotted versus this 01 in fig. 5. Once more, a sharp increase of the etch pit slope can be seen at a —0.6% ((110) dislocations, generation I etch pits) and at a 1 —0.85% ((100> dislocations, generation II etch pits). From the value of the critical undersaturation we can estimate the value of the edge free energy of a step y, using eq.0b(3). The on rough based fig. 5apis proximation a and too weak for between detailed calculations with a7 and we will use for a~’the value of 1%, which was derived from the rotating disc experiments (section 4.2). The edge free energy is not equal for steps in all directions, so we can only establish a mean value for y. The critical (interfacial) chemical potential difference np.” in eq. (3) is related to a7 by eq. (13). In table I, the parameters necessary for the determination of y are summarized. Bennema [29]estimates the edge free energy from growth rate versus supersaturation plot of —

etch pits, low dissolution rate and steep etch pits, high dissolution rate) can be distinguished, a!though the transition is not very sharp. One of the reasons is the different dislocations on the (II 1) face (b <110) or <100>), which have different ~ (see section 3). From fig. 10 we conclude that the critical surface undersaturation a is about 1%. The same conclusion was drawn from fig. 9, where at high undercoolings a surface undersaturation of 1% was attained. =

—

—

Table I Parameters of the <110) and K 100) dislocation types for the determination of edge energy y

5. Discussion Both from the surface observations and from the dissolution kinetics we conclude that the extended CL theory holds for the dissolution of the (Ill) face of potash alum. From the microtopographic investigations, we saw that the two dislocation types, <110) and (100), are marked by steep etch pits above two different critical bulk undersaturations of 0.8 and 1.8 K. From the dissolution kinetics we saw an enormous increase in the dissolution rate at a certain undersaturation. Withinterface the results of the rotatingcandisc the undersaturation be experiments determined after the elimination of the concentration jump due to the volume diffusion, which yielded a critical interfacial undersaturation of 1%. —

Parameter ~

(m3)

~i~/kT

—3.3X

IbKlOO)

bI
(m) (m)

K(~~)
K )L y

Value 9x 1028 a) — lOX 10-2 10-2

l.22X l0~ l.72x l0~ 118 1.0 303

Source Ref. [27] Experiment Experiment and eq. (13) Ref. [27] Ref [27] Ref. [28] Ref. [28] Experimental condition

(J/m3) 0.58x 10I0 Ref. [28] 2) 0.009—0.011 b) Eq. (3) (J/m . As one growth unit we consider one molecule K 2S04. A12(S04)3.24H20, but other growth unit sizes do not affect the result.

)~1This

value of y holds close to the dislocation centre.

B. van der Hock et a!.

/

Dissolution kinetics and etch pit studies of potash alum

193

potash alum. On the assumption that the growth rate is determined by monomolecular spiral hillocks, which grow according to the BCF theory [6], he finds y 0.0025 J/m2. We can also compare

[5] B. van der Hoek, J.P. van der Eerden, P. Bennema and I. Sunagawa, J. Crystal Growth 58 (1982) 365. [6] W.K. Burton, N. Cabrera and F.C. Frank, Phil. Trans. London 243 Symp. (1951) on 299.Internal Stress (Institute [7] Roy. W.L. Soc. Bragg, in: Proc. of Metals, London, 1947) p. 221. [8] V.G. Levich, Physicochemical Hydrodynamics (Prentice-

although the latter energy is larger than the former. Mullin and Zacek [30]estimate Ysurf from induc-

Hall, Englewood Cliffs, NJ, 1962). [9] D.P. Gregory and AC. Riddiford, J. Chem. Soc. (1956) 3756. [10] JR. Bourne, R.J. Davey, H. Gros and K. Hungerbtihler, J.

the value of the edge free energy with the reported values for the surface free energy of potash alum,

tion periods of homogeneous nucleation: “(surf 0.003 J/m2. The Nielsen—Sohnel empirical relationship [31], based on equilibrium solubility, predicts much higher values: ‘(surf 0.05 J/m2. How-

process is not entirely determined by volume diffusion. (ii) The extended CL dissolution theory holds

Crystal Growth 34 (1976) 221. [II] R. Janssen-van Rosmalen, C. van Leeuwen, G.K. Richtervan Leeuwen and J.M. Smith, Delft Progr. Rept., Ser. A, I (1976) 150. [12] R. Janssen-van Rosmalen, Thesis, Technical University Delft (1977). [13] J.W. Mullin, J. Garside and R. Unahabhoka, J. Appl. Chem. 15 (1965) 502. [14] S.J. Janëiá, Thesis, University of London (1976). [15] P.J. Reilly, RH. Wood and R.A. Robinson, J. Phys. Chem. 75 (1971) 1305. [16] F. Lenzi, Tuong-Tu Tran and Tjoon-Tow Teng, Can. J.

for both <100) and <110> dislocations ending on (111). (iii) The critical interfacial undersaturation is about — 1%. (iv) The edge free energy is about

Chem. 53(1975)3133. [17] R.A. Robinson and RH. Stokes, Electrolyte Solutions (Butterworths, London, 1955).

0.01 J/m2, or y/kT per growth unit is 2.2. (v) Growth of potash alum is determined by the surface processes.

[181 W.J.P. van Enckevort, P. Bennema and W.H. van der Linden, Z. Physik. Chem. NF 124 (1981) 171. [19] W.J.P. van Enckevort, Hi. Human and W.H. van der Linden, J. Crystal Growth, to be published. [20] H.J. Human, W.J.P. van Enckevort and W.H. van der Linden, J. Crystal Growth, to be published.

Acknowledgements

[21] W.J.P. van Enckevort and W.H. van der Linden, J. Crystal Growth 47 (1979) 196. [22] H. Komatsu, in: Crystal Growth and Characterisation, Eds. R. Ueda and J.B. Mullin (North-Holland, Amsterdam, 1975) p. 333. [23] AR. Lang, Acta Cryst. 12 (1959) 249. [24] 5. Gits-Leon, F. Lefaucheux and MC. Robert, J. Crystal

ever, in all previous references, concentrations — instead of activities — were used for the super(un-

der)saturation, which causes that the reported values are a factor of about two too low. The final conclusions are: (i) The dissolution

The authors are indebted to Ing. H.-J. Boeshaar for technical assistance and to Professor P. Bennema for stimulating discussions. Two of us (B. v.d. H. and W. v. E.) were supported by the Netherlands

Foundation

for

Pure

Research,

ZWO/SON. References [11 N. Cabrera and MM. Levine, Phil. Mag. 1(1956) 450. [2] W. Schaarwaechter, Phys. Status Solidi 12 (1965) 375. [3] W. Schaarwaechter, Phys. Status Solidi 12 (1965) 865. [4] B. van der Hoek, J.P. van der Eerden and P. Bennema, J. Crystal Growth 56 (1982) 621.

44 (1978) 345. and J.G.M. Odekerken, Phil. Mag., [25] Growth W.J.P. van Enckevort to be published. [26] H.J. Human, Thesis, Catholic University Nijmegen (1981). [27] Strukturbericht, Vol. III, re-edited by C. Gottfried and F. Stossberger (Akademie Verlang, Leipzig, 1937) p. 108. [28] S. Haussuehl, Fortschr. Mineral. 36 (1958) 75. [29] P. Bennema, R. Kern and B. Simon, Phys. Status Solidi 36 (1958) 75. [30] J.W. Mullin and S. Zá~ek,J. Crystal Growth 53 (1981) 515. [31] A.E. Nielsen and 0. Sohnel, J. Crystal Growth 11(1971) 233.