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Measurements of surface supersaturations around a growing K-alum crystal in aqueous solution
Journal of Crystal Growth 98 (1989) 377—383 North-Holland, Amsterdam
377
MEASUREMENTS OF SURFACE SUPERSATURATIONS AROUND A GROWING K-ALUM CRYSTAL IN AQUEOUS SOLUTION K. ONUMA, K. TSUKAMOTO and I. SUNAGAWA
*
Institute of Mineralogy, Petrology and Economic Geology, Faculty of Science, Tohoku University, Aoba, Sendai 980, Japan
Received 17 April 1989; manuscript received in final form 11 July 1989
Surface supersaturations, a,, were directly measured by the Mach—Zehnder interferometer on K-alum crystals growing in the 0b’ at aqueous solution, at different flow velocities u, from 3 to 40 cm/s. The ~ is not linearly related to the bulk supersaturation, smaller u, and is always smaller by 6 to 20% than a 5. The profile of ~ distribution over a growing face changes significantly and becomes more asymmetrical and shows a wider uniform a, region as increasing u.
1. Introduction Since Bennema [1—3]investigated the growth mechanism of crystals from aqueous solutions by the weighing method, extensive work has been carried out using a similar method. In these experiments, the bulk supersaturations ab were conveniently taken as representing the driving force, although the investigators know implicitly that the real driving force is the surface supersaturation;. This was simply because of the difficulties in measuring or evaluating the surface supersaturation. We should, however, note that there is no 0b~ guarantee that ;a, is to ; and Gb, To evaluate or linearly relationsrelated between we need to know the concentration gradient in the diffusion boundary layer. Bedarida [4] applied the multidirectional laser holographic technique to the growth of NaClO 3 and sucrose crystals and drew concentration maps around the growing crystals. Van Dam et al. [5] employed micro-Mach—Zehnder interferometry to measure the concentration difference around a growing perovskite crystal and demonstrated0hthat wasrecently a largeBedarida difference and there a,. Very et between the investigated the effects of the conal. [6] have
centration gradient upon the tapering of sucrose crystals using a holographic method. In all these experiments, growth was performed in a stagnant solution and the growth rates were not measured. Hirota et al. [7] measured the growth rate and the surface supersaturation of K-alum crystals growing in a running solution by means of the classical weighing technique and the Mach—Zehnder interferometry, respectively. However, both measurements were not simultaneously performed. The aim of this work is to realize simultaneous measurements of the growth rates and the surface supersaturations K-alum crystals at different flow rates u. Inof this paper, the experimental method developed for this purpose and the results obtained on versus Gb relations, two-dimensional a, distributions over a growing face, and their dependence upon the flow rates u are reported. The results and analysis of growth and dissolution kinetics in relation to a, will be reported in the forthcoming paper [8].
2. Experimental 2.1. Total set-up
*
Present address: Kashiwa-cho 3-54-2, Tachikawa, Tokyo 190, Japan.
The experimental set-up is schematically illustrated in fig. 1. The whole system is installed in
0022-0248/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
378
K. Onuma et al.
Video record.
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Surface supersaturations around growing K.alum crystal in aqueous solution
imer
image proc.
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Michelson interferorneter
~~,~~eftector
I—I ~ ~~G~wthCell
Mach-Zehnder inter erorneter
cube prism ~
~ reflector
beam splitter
beam prism expander
Fig. 1. Total experimental set-up. Simultaneous observations of growing surface of a crystal and concentration gradient in the environment by vertical and horizontal lights are realized for the first time by this set-up. Mach—Zehnder interferometry is applied for the horizontal observations, and Michelson interferometry and differential interference contrast microscopy (DICM) for the vertical observations.
a clean room with class 1000. This set-up consists of (a) growth cell connected with two-bath flow system, (b) optical system equipped with microMichelson interferometer for growth rate measurement and surface microtopographic observation, (c) optical system for Mach—Zehnder interferometry and (d) recording system. To perform such simultaneous measurements, observations from both vertical and horizontal directions are necessary. 2.2. Growth experiments Seed crystals of about 5 mm across preparedseparately were placed in the growth cell, and growth or dissolution was performed in forced flow of super- or undersaturated solution by the two-bath flow system. The temperature in the growth cell was controlled by computer with an accuracy of ±0.010 C. The bulk supersaturations are calculated based on the solubility curve determined by Jan~ié[9].
2.3. Measurement of growth rate of a spiral hillock A micro-Michelson interferometer which is the same as that used by Maiwa [10] was used to measure in a short time the normal growth rate, the slope and the step advancing rate of an mdividual growth hillock on the (111) face of K-alum crystals. A similar technique was also employed by Chernov et al. [11—13].This interferometer is attached to the objective lenses of a microscope with magnifications of 5 and 10 times. It is very compact and convenient to use (about 10 cm in length). A He—Ne laser was used as the light source due to its high coherency and thus the easiness for obtaining the interference fringes. 2.4. Measurement of surface supersaturation A Mach—Zehnder interferometer was used to directly measure the a, and their two-dimensional distributions over a face while the growth rate etc. were being measured. The same He—Ne laser as
K. Onuma et at.
/ Surface
379
supersaturations around growing K-alum crystal in aqueous solution
for the Michelson interferometer is employed as the light source. The difference of concentrations between the bulk and the crystal surface appears as a bend of the interference fnnges. Thus the surface concentration and the thickness of concentration boundary layer are easily determined in situ by measuring the shift of fringes, provided that the relation between refractive indices n and concentration is known.
Rne~axctive
,\ (constant Cone.) ~ n~
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3. Results
!
3.1. Refractive indices and a~
To determine the a~from the shift of interference fringes, it is necessary to know both the dependence of the refractive index upon the ternperature under a constant concentration and the refractive indices at each saturation temperatures. The former is measured as follows. K-alum solution which is saturated at 35.50°C was put into the growth cell described in our previous paper [14]. The temperature was changed slowly, and the shift of the Mach—Zehnder interferometric fringes was followed on a TV screen. This method is very sensitive and the order of 10 —5 —10 —6 difference of refractive index is detec.
.
.
table. The result is shown by the solid line in fig. 2. Since the absolute values of the refractive indices at different saturation temperatures are not obtainable by the above method, we adopted the data measured by Takubo et al. [15] by means of modified Reyners refractometer, after converting the values to fit He—Ne laser wavelength. The dashed line in fig. 2 indicates the data. In fig. 2, b’ n5, and ~ n are the refractive indices of bulk and surface solutions, and the difference, respectively. ~n, and thus n5, is calculated from the amount of fringe shift. Since n5 at a certain ternperature is determined, a parallel straight line passing throughThis the is point (openthat circle) the solid line is drawn. because the toconcentration difference between the bulk and crystal surface is very small, and thus the dependence of n on the temperature does not change significantly. The point where this line intersects the
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Temp.(°C) Fig. 2. The refractive index of saturated K-alum solutions (dashed line) and the dependence of refractive index upon the temperature under a constant concentration (solid line). ~ n indicates the difference of refractive indices of solutions between bulk, n 5, and crystal surface, n . T is the saturation temperature.
dashed line corresponds to the concentration of the solution saturated at T0. Once the temperature 7~is determined, the absolute concentration at the growing surface is easily derived from the solubility curve, and a~is calculated. 3.2. Variation of
G,
along a growing face
Two-dimensional distributions of a5 over a growing (111 } face are measured at different flow velocities, u on3, the 10, same 40 cm/s, 0b and crystal.but under a constant 3.2.1. u 3 cm/s Fig. 3 shows the result obtained at u 3 cm/s. The left vertical axis represents the a, value, the right axis the thickness of concentration boundary =
=
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Fig. 3. Two-dimensional distribution of a, at u = 3 cm/s under the condition of constant a~.Solid circles are the data of a,, closed circles are the thickness of the concentration boundary layer, and the reverse closed triangle shows the position of the only growth hillock present on the { 111) face.
layer, and the abscissa the distance from the front edge of the crystal facing to the flow direction. The reverse solid triangle on the abscissa indicates the position of a growth hillock which is the only one present on the whole surface. a 5 is 4.50% near the front edge of the crystal, which is the detectable limit from the shift of interference fringes. The ; value decreases gently when going to the center of the face and reaches the minimum value, 4.09%, at about 3 mm inside from the edge. ; is ca. 20% lower than Gb. When approaching the rear edge of the crystal, a5 again increases slightly, The two-dimensional profile of a~over the face is more or less symmetrical, and the region with nearly constant a~value is rather narrow, limited only around the center of the surface. Closed triangles represent the measured ~ values at each point. The open circles show the values calculated theoretically from the equation of the thickness of boundary layer. When the flow is laminar along the infinite length of a flat plane, the thickness of the boundary layer, is expressed by the diffusion constant D, the kinematic ~,
~,
viscosity v, the distance from the front edge x, and the flow velocity u as ~
=
C(D/v)5~’3(vx/u)
1/2
The values of D and v were determined by Mullin et al. [16]. The constant C changes when the fluid changes. Using the values of D and i’, l~ is plotted for C 2 and 3.5, respectively. The measured values fit well the curve of C 3.5 up to 3 m.m from the front edge. However, the values gradually deviate more from the theoretical curve as x increases and they fit better the curve of C 2. =
=
=
3.2.2. u 10 cm/s Fig. 4 shows the result obtained for u 10 cm/s. Gb is 4.88%, which is slightly smaller than that in the case of section 3.2.1, but the difference is too small to assume any significant effect to the results. The same crystal was used, on which only one growth hillock presents on the surface, although its position is slightly different. =
=
K. Onuma et a!.
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Surface supersaturations around growing K-alum crystal in aqueous solution
381
thickness of boundary layer
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The two-dimensional profile of a~is clearly different from that of u 3 cm/s. It shows an asymmetric form and ; decreases more sharply when leaving from the front edge and drops down to 4.33% at only 2 mm inside. The drop of; from
the front edge to this point is large, but after this drop, a~does not change significantly over a wide distance up to the rear edge. There is a wide constant ; range over the surface, where a5 is 13% lower than the Gb.
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growth3 hillock Fig. 5. Two-dimensional distribution of a, at the highest flow velocity, u
4 =
I
5 (mm)
40 cm/s. Symbols are the same as in fig. 3.
382
K. Onuma et a!.
/
Surface supersaturations around growing K-alum crystal in aqueous solution
The thickness of the concentration boundary
G
5(’.ol o,io (°/,)
layer fits well the theoretical curve of C 3.5, similar to the case of section 3.2.1, although it shows a lower value at 4 mm to 4.5 mm away from the front edge. 3.2.3. u
=
40 cm/s
~
8
6 4
The result at u 40 cm/s is shown in fig. 5. This is the maximum flow velocity in the present experiments. The same crystal was used and the number of growth hillocks is also the same. The two-dimensional profile of a5 is, however, different from the case of section 3.2.2. a~becomes constant at a distance of 700—800 ~tm from the front edge, and a very wide range of constant a~is realized. No serious discrepancy from this value could be observed, at least within a detectable range of fringe shift. The a, distribution over the face is asymmetrical. All these features suggest that stable and uniform growth is realized over a wide area, except the narrow regions at both ends of the crystal. a, near the center of the face is 4.59% is 6% ofsmaller than Gb. The which thickness the concentration boundary layer does not fit any theoretical curves, and is practically constant over the whole surface. This suggests that the flow pattern does not coincide with the one which is assumed for the laminar flow along the infinite length plane at higher flow velocity,
0,13)
surloce
=
5
=
3
s
3cm/s
A
10cm/s
•
4Ocm/s
6 .
“ ,,,,.
-
3 2 2 2 1 1
o
~ersoturat~on I
o
2 plot at u 4= 3, 10, 40 6cm/s. The 8three Fig. 6. a,axes versus vertical from right to left correspond to the cases of 3 cm/s (a,(3)), 10 cm/s (a,(10)), and 40 cm/s (a,(40)), respectively. Note the nonlinear relation at u = 3 cm/s and the linear relation at u = 10 and 40 cm/s. 0h
increases. The fitting curve assuming a continuous variation of a~with Gb is a 92.However, 5 line 0.95 upa~to about 3% a,, seems to lie on a straight Gb, and above this, a~changes nonlinearly. The fitting curves in this case are a~ 0.88 ab (ab ~ 3%) and a.~ 0.99 a~59(Gb> 3%). The dashed line is an extension of the linear relation in the lower Gb region. The match between the curves and the =
3.3. Surface supersaturation versus bulk supersaturation
=
=
The a, measured in relation to the Gb at three flow velocities, u 3, 10 and 40 cm/s, are shown in fig. 6. The measurements of a 5 were made at the central portion of a face where uniform a~can be assumed, and the obtained data were averaged, =
measured values is better than in the former case. a, is at the maximum about 20% lower than ab within the experimental range.
3.3.1. u = 3 cm/s
3.3.2. u
The right vertical axis indicated by a5 (3) in fig. 6 indicates the a, value which corresponds to this flow velocity and closed circles are the measured values. It is clearly seen that a, does not change 0b~The ; values gradulinearly in relation ally deviate from to thethestraight line and the difference between a, and ab is pronounced as °b
The second left vertical axis indicated by a, (10) corresponds to the case for u 10 cm/s. The closed triangles are the measured values, which are obviously different from curve in the of section 3.3.1. Namely, the the dependence of a,case upon Gb is linear in this case. The curve fits very well the line a, 0.87 ab obtained by the least squares
=
10 cm/s =
=
K. Onuma et a!.
/ Surface supersaturations
method. a5 in this case is about 13% lower than over the whole range of the experiment.
383
around growing K-alum crystal in aqueous solution
Gb
(2) The a5 versus ab relations are not linear at a small flow velocity, u 3 cm/s, but become linear at a larger flow velocity than u 10 cm/s. a, is about 20% lower at maximum than Gb within the experimental range, which implies that the surface growth kinetics resistance is stronger than the mass transport resistance. =
=
3.3.3. u 40 cm/s The left vertical axis indicated by a,(40) shows a, at the highest flow velocity, u 40 cm/s. The relation between a, and Gb is also linear similarly as in the case of u 10 cm/s and the data indicated by closed squares are fitted by the line with 0.94 Gb. a, is about 6—6.5% smaller than Gb. Although the difference is small, it can not be neglected because Gb is controlled with less accuracy than 0.03% which is derived from the fluctuation of the solution temperature. As the difference between a5 and Gb reaches to the critical value below 1.5% Gb, only the data above this ab are shown in the figure and the obtained line is extrapolated to the region below 1.5% Gb. =
=
=
=
Acknowledgements The authors thank Professor H. Takubo of Osaka University for providing us with the unpublished refractive index data of K-alum solutions. This work is supported by a Grant-in-Aid for fundamental research from the Japan Ministry of Education, Science and Culture.
References
4. Swnmary and discussion
The results are summarized as follows. (1) The profile of two-dimensional distributions of a5 over a face varies significantly depending upon the flow velocities, u. As increasing u, the profile changes from a symmetrical to an asymmetrical one, and the region of nearly constant a, becomes wider. The thickness of concentration boundary layer, deviates from the theoretical curve as u increases. When the flow velocity is small, the shape of the concentration boundary layer takes nearly the same form as in a stagnant solution. Since the flow arrival at the front face may take an easy roundabout to the side face in this case, giving no severe effect to the boundary layer, and thus the profile is symmetrical. This makes a, higher at both front and rear edges than at the central area. When u is increases, an easy roundabout becomes no more possible, and the flow moves along the vertical side face to the top horizontal face, which results in a steeper boundary layer at the front edge. Asymmetry of the profile is thus enhanced as u increases. ~,
[1] P. Bennema, Phys. Status Solidi 17 (1966) 563. [2] P. Bennema, J. Crystal Growth 1 (1967) 278. [31P. Bennema, J. Crystal Growth 1 (1967) 287. [4] F. Bedarida, J. Crystal Growth 79 (1986) 43. [5] J.C. van Dam and F.H. Mischgofsky, J. Crystal Growth 539. L. Zefiro, P. Boccacci, D. Aquilano. M. [6] 84 F. (1987) Bedarida, Rubbo, G. Vaccari, G. Mantovani and G. Sgualdino, J. Crystal Growth 89 (1988) 395. [7] S. Hirota, K. Fukui and M. Nakajima, Kagaku Kogyo Ronbunsyu 2 (1976) 552 (in Japanese). K. Onuma, K. Tsukamoto and I. Sunagawa, J. Crystal Growth, in press. S.J. Jan~iá,Thesis, University of London (1976). K. Maiwa, PhD Thesis, Tohoku University (1987). A.A. Chernov, L.N. Rashkovich and A.A. Mkrtchan, J. Crystal Growth 74 (1986) 101. [12] A.A. Chernov, L.N. Rashkovich and A.A. Mkrtchan, Soviet Phys.-Cryst. 32 (1987) 432. [13] A.A. Chernov and L.N. Rashkovich, J. Crystal Growth 84 (1987) 389. [14] K. Onuma, K. Tsukamoto and I. Sunagawa, J. Crystal [8] [9] [10] [11]
Growth 89 (1988) 177. [15] H. Takubo, Abstract, Annual Meeting Mineralogical Society of Japan, 1987; H. Takubo, private communication. [16] J.W. Mullin, J. Garside and R. Unahabhokha, J. AppI. Chem. 15 (1965) 502.
377
MEASUREMENTS OF SURFACE SUPERSATURATIONS AROUND A GROWING K-ALUM CRYSTAL IN AQUEOUS SOLUTION K. ONUMA, K. TSUKAMOTO and I. SUNAGAWA
*
Institute of Mineralogy, Petrology and Economic Geology, Faculty of Science, Tohoku University, Aoba, Sendai 980, Japan
Received 17 April 1989; manuscript received in final form 11 July 1989
Surface supersaturations, a,, were directly measured by the Mach—Zehnder interferometer on K-alum crystals growing in the 0b’ at aqueous solution, at different flow velocities u, from 3 to 40 cm/s. The ~ is not linearly related to the bulk supersaturation, smaller u, and is always smaller by 6 to 20% than a 5. The profile of ~ distribution over a growing face changes significantly and becomes more asymmetrical and shows a wider uniform a, region as increasing u.
1. Introduction Since Bennema [1—3]investigated the growth mechanism of crystals from aqueous solutions by the weighing method, extensive work has been carried out using a similar method. In these experiments, the bulk supersaturations ab were conveniently taken as representing the driving force, although the investigators know implicitly that the real driving force is the surface supersaturation;. This was simply because of the difficulties in measuring or evaluating the surface supersaturation. We should, however, note that there is no 0b~ guarantee that ;a, is to ; and Gb, To evaluate or linearly relationsrelated between we need to know the concentration gradient in the diffusion boundary layer. Bedarida [4] applied the multidirectional laser holographic technique to the growth of NaClO 3 and sucrose crystals and drew concentration maps around the growing crystals. Van Dam et al. [5] employed micro-Mach—Zehnder interferometry to measure the concentration difference around a growing perovskite crystal and demonstrated0hthat wasrecently a largeBedarida difference and there a,. Very et between the investigated the effects of the conal. [6] have
centration gradient upon the tapering of sucrose crystals using a holographic method. In all these experiments, growth was performed in a stagnant solution and the growth rates were not measured. Hirota et al. [7] measured the growth rate and the surface supersaturation of K-alum crystals growing in a running solution by means of the classical weighing technique and the Mach—Zehnder interferometry, respectively. However, both measurements were not simultaneously performed. The aim of this work is to realize simultaneous measurements of the growth rates and the surface supersaturations K-alum crystals at different flow rates u. Inof this paper, the experimental method developed for this purpose and the results obtained on versus Gb relations, two-dimensional a, distributions over a growing face, and their dependence upon the flow rates u are reported. The results and analysis of growth and dissolution kinetics in relation to a, will be reported in the forthcoming paper [8].
2. Experimental 2.1. Total set-up
*
Present address: Kashiwa-cho 3-54-2, Tachikawa, Tokyo 190, Japan.
The experimental set-up is schematically illustrated in fig. 1. The whole system is installed in
0022-0248/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
378
K. Onuma et al.
Video record.
,-
/
Surface supersaturations around growing K.alum crystal in aqueous solution
imer
image proc.
C.) C)
DICM
Michelson interferorneter
~~,~~eftector
I—I ~ ~~G~wthCell
Mach-Zehnder inter erorneter
cube prism ~
~ reflector
beam splitter
beam prism expander
Fig. 1. Total experimental set-up. Simultaneous observations of growing surface of a crystal and concentration gradient in the environment by vertical and horizontal lights are realized for the first time by this set-up. Mach—Zehnder interferometry is applied for the horizontal observations, and Michelson interferometry and differential interference contrast microscopy (DICM) for the vertical observations.
a clean room with class 1000. This set-up consists of (a) growth cell connected with two-bath flow system, (b) optical system equipped with microMichelson interferometer for growth rate measurement and surface microtopographic observation, (c) optical system for Mach—Zehnder interferometry and (d) recording system. To perform such simultaneous measurements, observations from both vertical and horizontal directions are necessary. 2.2. Growth experiments Seed crystals of about 5 mm across preparedseparately were placed in the growth cell, and growth or dissolution was performed in forced flow of super- or undersaturated solution by the two-bath flow system. The temperature in the growth cell was controlled by computer with an accuracy of ±0.010 C. The bulk supersaturations are calculated based on the solubility curve determined by Jan~ié[9].
2.3. Measurement of growth rate of a spiral hillock A micro-Michelson interferometer which is the same as that used by Maiwa [10] was used to measure in a short time the normal growth rate, the slope and the step advancing rate of an mdividual growth hillock on the (111) face of K-alum crystals. A similar technique was also employed by Chernov et al. [11—13].This interferometer is attached to the objective lenses of a microscope with magnifications of 5 and 10 times. It is very compact and convenient to use (about 10 cm in length). A He—Ne laser was used as the light source due to its high coherency and thus the easiness for obtaining the interference fringes. 2.4. Measurement of surface supersaturation A Mach—Zehnder interferometer was used to directly measure the a, and their two-dimensional distributions over a face while the growth rate etc. were being measured. The same He—Ne laser as
K. Onuma et at.
/ Surface
379
supersaturations around growing K-alum crystal in aqueous solution
for the Michelson interferometer is employed as the light source. The difference of concentrations between the bulk and the crystal surface appears as a bend of the interference fnnges. Thus the surface concentration and the thickness of concentration boundary layer are easily determined in situ by measuring the shift of fringes, provided that the relation between refractive indices n and concentration is known.
Rne~axctive
,\ (constant Cone.) ~ n~
ns 1.3465
-
/
3. Results
!
3.1. Refractive indices and a~
To determine the a~from the shift of interference fringes, it is necessary to know both the dependence of the refractive index upon the ternperature under a constant concentration and the refractive indices at each saturation temperatures. The former is measured as follows. K-alum solution which is saturated at 35.50°C was put into the growth cell described in our previous paper [14]. The temperature was changed slowly, and the shift of the Mach—Zehnder interferometric fringes was followed on a TV screen. This method is very sensitive and the order of 10 —5 —10 —6 difference of refractive index is detec.
.
.
table. The result is shown by the solid line in fig. 2. Since the absolute values of the refractive indices at different saturation temperatures are not obtainable by the above method, we adopted the data measured by Takubo et al. [15] by means of modified Reyners refractometer, after converting the values to fit He—Ne laser wavelength. The dashed line in fig. 2 indicates the data. In fig. 2, b’ n5, and ~ n are the refractive indices of bulk and surface solutions, and the difference, respectively. ~n, and thus n5, is calculated from the amount of fringe shift. Since n5 at a certain ternperature is determined, a parallel straight line passing throughThis the is point (openthat circle) the solid line is drawn. because the toconcentration difference between the bulk and crystal surface is very small, and thus the dependence of n on the temperature does not change significantly. The point where this line intersects the
/ To.;
/ 1.3460 I
31
32
33
Ts
j
34
35
36
37
Temp.(°C) Fig. 2. The refractive index of saturated K-alum solutions (dashed line) and the dependence of refractive index upon the temperature under a constant concentration (solid line). ~ n indicates the difference of refractive indices of solutions between bulk, n 5, and crystal surface, n . T is the saturation temperature.
dashed line corresponds to the concentration of the solution saturated at T0. Once the temperature 7~is determined, the absolute concentration at the growing surface is easily derived from the solubility curve, and a~is calculated. 3.2. Variation of
G,
along a growing face
Two-dimensional distributions of a5 over a growing (111 } face are measured at different flow velocities, u on3, the 10, same 40 cm/s, 0b and crystal.but under a constant 3.2.1. u 3 cm/s Fig. 3 shows the result obtained at u 3 cm/s. The left vertical axis represents the a, value, the right axis the thickness of concentration boundary =
=
=
380
/
K Onuma et at.
Surface supersaturations around growing K-alum crystal in aqueous solution
Gs(°/o)
thick~e~~ of boundary layer
u:3cm,~s. 4.900/o
(P.m)
Ob=
4.5.
4.0
•
-
0~~~~~
100 o~o A
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3.5 LA £
C’2
50
£
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0
Fig. 3. Two-dimensional distribution of a, at u = 3 cm/s under the condition of constant a~.Solid circles are the data of a,, closed circles are the thickness of the concentration boundary layer, and the reverse closed triangle shows the position of the only growth hillock present on the { 111) face.
layer, and the abscissa the distance from the front edge of the crystal facing to the flow direction. The reverse solid triangle on the abscissa indicates the position of a growth hillock which is the only one present on the whole surface. a 5 is 4.50% near the front edge of the crystal, which is the detectable limit from the shift of interference fringes. The ; value decreases gently when going to the center of the face and reaches the minimum value, 4.09%, at about 3 mm inside from the edge. ; is ca. 20% lower than Gb. When approaching the rear edge of the crystal, a5 again increases slightly, The two-dimensional profile of a~over the face is more or less symmetrical, and the region with nearly constant a~value is rather narrow, limited only around the center of the surface. Closed triangles represent the measured ~ values at each point. The open circles show the values calculated theoretically from the equation of the thickness of boundary layer. When the flow is laminar along the infinite length of a flat plane, the thickness of the boundary layer, is expressed by the diffusion constant D, the kinematic ~,
~,
viscosity v, the distance from the front edge x, and the flow velocity u as ~
=
C(D/v)5~’3(vx/u)
1/2
The values of D and v were determined by Mullin et al. [16]. The constant C changes when the fluid changes. Using the values of D and i’, l~ is plotted for C 2 and 3.5, respectively. The measured values fit well the curve of C 3.5 up to 3 m.m from the front edge. However, the values gradually deviate more from the theoretical curve as x increases and they fit better the curve of C 2. =
=
=
3.2.2. u 10 cm/s Fig. 4 shows the result obtained for u 10 cm/s. Gb is 4.88%, which is slightly smaller than that in the case of section 3.2.1, but the difference is too small to assume any significant effect to the results. The same crystal was used, on which only one growth hillock presents on the surface, although its position is slightly different. =
=
K. Onuma et a!.
/
Surface supersaturations around growing K-alum crystal in aqueous solution
381
thickness of boundary layer
Os(°/o)
u~10cmis.
(~.Lm)
Gb4.88°/o
4.5
~
S
~
100
S 0~~~~ ~~~~~~-0
0~~~~~
o~ A
A
A
A
_,, 4,~I0_...-.A
4.0~
0
C’3.5
___________
,.....0
1
2
growth
£
50 0
A
I
0 I
I
3
4
5 (mm)
C’ 2
hiltock Fig. 4. Two-dimensional distribution of a, at u =10 cm/s. Symbols are the same as in fig. 3.
The two-dimensional profile of a~is clearly different from that of u 3 cm/s. It shows an asymmetric form and ; decreases more sharply when leaving from the front edge and drops down to 4.33% at only 2 mm inside. The drop of; from
the front edge to this point is large, but after this drop, a~does not change significantly over a wide distance up to the rear edge. There is a wide constant ; range over the surface, where a5 is 13% lower than the Gb.
=
thickness
Os(°/o)
u
5.0
boundary layer
Of
40cm/s 4.BB°/o
Qim)
Ob~
p
4.5
5
p
•
pp•
_S
—
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0~C’5 A
4.0
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A
o— C’35
A 0
~
20
o~ V
1
.
0
o~
0
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I
I
2
growth3 hillock Fig. 5. Two-dimensional distribution of a, at the highest flow velocity, u
4 =
I
5 (mm)
40 cm/s. Symbols are the same as in fig. 3.
382
K. Onuma et a!.
/
Surface supersaturations around growing K-alum crystal in aqueous solution
The thickness of the concentration boundary
G
5(’.ol o,io (°/,)
layer fits well the theoretical curve of C 3.5, similar to the case of section 3.2.1, although it shows a lower value at 4 mm to 4.5 mm away from the front edge. 3.2.3. u
=
40 cm/s
~
8
6 4
The result at u 40 cm/s is shown in fig. 5. This is the maximum flow velocity in the present experiments. The same crystal was used and the number of growth hillocks is also the same. The two-dimensional profile of a5 is, however, different from the case of section 3.2.2. a~becomes constant at a distance of 700—800 ~tm from the front edge, and a very wide range of constant a~is realized. No serious discrepancy from this value could be observed, at least within a detectable range of fringe shift. The a, distribution over the face is asymmetrical. All these features suggest that stable and uniform growth is realized over a wide area, except the narrow regions at both ends of the crystal. a, near the center of the face is 4.59% is 6% ofsmaller than Gb. The which thickness the concentration boundary layer does not fit any theoretical curves, and is practically constant over the whole surface. This suggests that the flow pattern does not coincide with the one which is assumed for the laminar flow along the infinite length plane at higher flow velocity,
0,13)
surloce
=
5
=
3
s
3cm/s
A
10cm/s
•
4Ocm/s
6 .
“ ,,,,.
-
3 2 2 2 1 1
o
~ersoturat~on I
o
2 plot at u 4= 3, 10, 40 6cm/s. The 8three Fig. 6. a,axes versus vertical from right to left correspond to the cases of 3 cm/s (a,(3)), 10 cm/s (a,(10)), and 40 cm/s (a,(40)), respectively. Note the nonlinear relation at u = 3 cm/s and the linear relation at u = 10 and 40 cm/s. 0h
increases. The fitting curve assuming a continuous variation of a~with Gb is a 92.However, 5 line 0.95 upa~to about 3% a,, seems to lie on a straight Gb, and above this, a~changes nonlinearly. The fitting curves in this case are a~ 0.88 ab (ab ~ 3%) and a.~ 0.99 a~59(Gb> 3%). The dashed line is an extension of the linear relation in the lower Gb region. The match between the curves and the =
3.3. Surface supersaturation versus bulk supersaturation
=
=
The a, measured in relation to the Gb at three flow velocities, u 3, 10 and 40 cm/s, are shown in fig. 6. The measurements of a 5 were made at the central portion of a face where uniform a~can be assumed, and the obtained data were averaged, =
measured values is better than in the former case. a, is at the maximum about 20% lower than ab within the experimental range.
3.3.1. u = 3 cm/s
3.3.2. u
The right vertical axis indicated by a5 (3) in fig. 6 indicates the a, value which corresponds to this flow velocity and closed circles are the measured values. It is clearly seen that a, does not change 0b~The ; values gradulinearly in relation ally deviate from to thethestraight line and the difference between a, and ab is pronounced as °b
The second left vertical axis indicated by a, (10) corresponds to the case for u 10 cm/s. The closed triangles are the measured values, which are obviously different from curve in the of section 3.3.1. Namely, the the dependence of a,case upon Gb is linear in this case. The curve fits very well the line a, 0.87 ab obtained by the least squares
=
10 cm/s =
=
K. Onuma et a!.
/ Surface supersaturations
method. a5 in this case is about 13% lower than over the whole range of the experiment.
383
around growing K-alum crystal in aqueous solution
Gb
(2) The a5 versus ab relations are not linear at a small flow velocity, u 3 cm/s, but become linear at a larger flow velocity than u 10 cm/s. a, is about 20% lower at maximum than Gb within the experimental range, which implies that the surface growth kinetics resistance is stronger than the mass transport resistance. =
=
3.3.3. u 40 cm/s The left vertical axis indicated by a,(40) shows a, at the highest flow velocity, u 40 cm/s. The relation between a, and Gb is also linear similarly as in the case of u 10 cm/s and the data indicated by closed squares are fitted by the line with 0.94 Gb. a, is about 6—6.5% smaller than Gb. Although the difference is small, it can not be neglected because Gb is controlled with less accuracy than 0.03% which is derived from the fluctuation of the solution temperature. As the difference between a5 and Gb reaches to the critical value below 1.5% Gb, only the data above this ab are shown in the figure and the obtained line is extrapolated to the region below 1.5% Gb. =
=
=
=
Acknowledgements The authors thank Professor H. Takubo of Osaka University for providing us with the unpublished refractive index data of K-alum solutions. This work is supported by a Grant-in-Aid for fundamental research from the Japan Ministry of Education, Science and Culture.
References
4. Swnmary and discussion
The results are summarized as follows. (1) The profile of two-dimensional distributions of a5 over a face varies significantly depending upon the flow velocities, u. As increasing u, the profile changes from a symmetrical to an asymmetrical one, and the region of nearly constant a, becomes wider. The thickness of concentration boundary layer, deviates from the theoretical curve as u increases. When the flow velocity is small, the shape of the concentration boundary layer takes nearly the same form as in a stagnant solution. Since the flow arrival at the front face may take an easy roundabout to the side face in this case, giving no severe effect to the boundary layer, and thus the profile is symmetrical. This makes a, higher at both front and rear edges than at the central area. When u is increases, an easy roundabout becomes no more possible, and the flow moves along the vertical side face to the top horizontal face, which results in a steeper boundary layer at the front edge. Asymmetry of the profile is thus enhanced as u increases. ~,
[1] P. Bennema, Phys. Status Solidi 17 (1966) 563. [2] P. Bennema, J. Crystal Growth 1 (1967) 278. [31P. Bennema, J. Crystal Growth 1 (1967) 287. [4] F. Bedarida, J. Crystal Growth 79 (1986) 43. [5] J.C. van Dam and F.H. Mischgofsky, J. Crystal Growth 539. L. Zefiro, P. Boccacci, D. Aquilano. M. [6] 84 F. (1987) Bedarida, Rubbo, G. Vaccari, G. Mantovani and G. Sgualdino, J. Crystal Growth 89 (1988) 395. [7] S. Hirota, K. Fukui and M. Nakajima, Kagaku Kogyo Ronbunsyu 2 (1976) 552 (in Japanese). K. Onuma, K. Tsukamoto and I. Sunagawa, J. Crystal Growth, in press. S.J. Jan~iá,Thesis, University of London (1976). K. Maiwa, PhD Thesis, Tohoku University (1987). A.A. Chernov, L.N. Rashkovich and A.A. Mkrtchan, J. Crystal Growth 74 (1986) 101. [12] A.A. Chernov, L.N. Rashkovich and A.A. Mkrtchan, Soviet Phys.-Cryst. 32 (1987) 432. [13] A.A. Chernov and L.N. Rashkovich, J. Crystal Growth 84 (1987) 389. [14] K. Onuma, K. Tsukamoto and I. Sunagawa, J. Crystal [8] [9] [10] [11]
Growth 89 (1988) 177. [15] H. Takubo, Abstract, Annual Meeting Mineralogical Society of Japan, 1987; H. Takubo, private communication. [16] J.W. Mullin, J. Garside and R. Unahabhokha, J. AppI. Chem. 15 (1965) 502.