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Quantitative studies of solute boundary layers around crystals by holographic phase-contrast interferometric microphotography
690
Journal of Crystal Growth 106 (1990) 690—694 North-Holland
QUANTITATIVE STUDIES OF SOLUTE BOUNDARY LAYERS AROUND CRYSTALS BY HOLOGRAPHIC PHASE-CONTRAST INTERFEROMETRIC MJCROPHOTOGRAPHY
*
YU Xiling, YUE Xuefeng, GAO Hangjun and CHEN Huanchu institute of Crystal Materials, Shandong University. Jinan 250100, People’s Rep. of China Received 24 March 1989: manuscript received in final form 2 June 1990
This paper discusses an effective real-time measuring technique — holographic phase-contrast interferometric microphotography — for the studies of solute boundary layers around a crystal and the essential conditions for precise measurements under complicated conditions. The experiments demonstrated that the thickness of the solute boundary layer increased linearly with increasing supersaturation in the free-convection State while the solute distribution within the boundar~ layer exhibited nonlinear variations.
1. Introduction
2. Holographic phase-contrast interferometric microphotography
During solution crystal growth processes, at the place where solid and liquid meet, the true behaviour of solute transport remains a subject of research. In recent years, some papers reported studies on the distribution of concentration field and measurements of the thickness of boundary layers by holographic interferometry [1—4].Unfortunately, the analyses reported in these papers were performed under free-convection conditions, and they presented neither the solute distribution within the boundary layer which is an important region for solute transport, nor the regularities of variations in the boundary layers under forced convection. In addition, the accuracy of the measuring methods was not discussed in these papers. This paper presents an effective real-time measuring technique holographic phase-contrast interferometric microphotography for the measurement of solute boundary layers and reports the essential conditions needed for carrying out precise measurements. Using this technique, the authors have measured the thickness of the boundary layers and the concentration distributions within the layers under different convection conditions. —
—
*
This research project is supported by the National Nature Science Foundation of China (NSFC).
0022-0248/90/$03.50 © ~990
—
2.1. Fundamental principle
The principle of holographic phase-contrast microphotography for crystal growth research has been discribed in a previous paper [5]. Combining this principle with that of holographic interferometry, the holographic phase-contrast interferometnc microphotography can be formed, to carry out a comparison of the filtered object waves at two different time. The crystal growth causes the phase variations of the light. When the phase difference satisfies the following formula, (2K+ 1)~, (I) ~
where IK~ O~ 1, 2 interference dark fringes are formed. Placing the interference patterns under the microscope, we can observe and calculate the particular details concerned. =
2.2. The essential conditions for carrying out precise measurements on solute boundary layers
In order to study the environmental phase of a special interface, this environmental phase has to be exposed fully. The simple and reliable way is to make the crystal face parallel to the optical axis of the subject beam.
Elsevier Science Publishers B.V. (North-Holland)
Yu Xiiing et ai.
/
Studies ofsolute boundary layers around crystals
Furthermore, when the incident light radiates upon the edge of any object, the diffraction fringes will appear. These diffraction fringes can be seen clearly in the phase-contrast microscope. Only when the measuring screen is placed on the image plane, are the number and width of the diffraction fringes of the crystal edge the least and the smallest, and is at the same time, the clarity of the morphological image of the crystal optimal. Therefore, the measuring screen can be precisely adjusted and located by using the edge diffraction in the phase-contrast micrography as a criterion, In the meantime, this technique can also be useful in calibrating whether the crystal face to be measured is parallel to the optical axis. Finally, if the size of the crystal is large or the supersaturation of the growing solution is high, attention must be paid to the influence of the refraction of the light and the edge effect. It is supposed in the interference equation that the light travels straightly; in fact, when the light passes through a region where the gradient of refractive index is high, the deviation caused by the refraction cannot be neglected. On the other hand, the light passes through the two ends of the boundary layer and this is a non-two-dimensional problem, so that corrections of the edge deviations
691
should be made in order to obtain a precise conelusion [6].
3. The measurements on solute boundary layer 3.1. Experiment
The light source used is a He—Ne laser. The temperature in the crystallizer is maintained at an accuracy of ±0.05°C by a super-thermostat. The volume of the solution is 250 ml. The solution was forced to convect by the electromagnetic agitator. The crystallite seed is KDP with a size of about 6 X 6 x 6 mm. A single-exposure hologram was made to the saturated solution and processed in situ; then a crystallite seed was put into the solution, carefully adjusted and located with the aid of phase-contrast micrography. Through the changing of ternperatures, different holograms of different boundary layer variations can be obtained (see fig. 1). If the convection state of solution changed under the same supersaturation, holograms of the changed boundary layer under free convection and forced convection will be obtained (see fig. 2). The variation phenomena shown above were re-
—
—
Fig. I. Holograms of different boundary layers for different supersaturations of KDP solution: (a) in crystal growth process = 1.04%; (b) in dissolving state ~C = —0.19%. Magnification 12 X.
692
Yu Xiling et aL
/ Studies ofsolute
boundary laveri around crystals
Fig. 2. Holograms of different convection states at a temperature of 29.05°c. .~( Magnification 12 x.
corded by microphotography. After processing the films, measurements and calculations were made, 3.2. Results and discussion 3.2.1. Thickness of solute boundary layer
The thickness ~ of the solute boundary layer changes with the variations of the solution supersatuation under free convection. The experimental results are shown in fig. 3, from which we can see that 8~increases linearly with an increase of the supersatuation. This experimental result is funda-
4~
/
/
/
300
/
250
200
/
//
mentally in agreement with the diffusion equation of Nernst. The thickness 8 of the Nernst diffusion layer is in proportion to the supersatuation directly. From this point of view, the solute boundary layer is the diffusion layer. However, the experiment also shows that the thickness of the boundary layer is not equal on the different faces of a crystal and at different positions of a crystal face. As shown in fig. 2a, the thickness 8~of the boundary layer at a pyramidal face of 5 mm distance to [010] is 99.3 ~tm, and 6~at a prismatic face of the same distance to [010] is 230.9 p~m.However, the 8~at the prismatic face of 10 mm distance to [010] is 286.8 !.Lm. This is not in agreement with the Nernst diffusion equation in which 8 should be equal in value at the same prismatic face of equal area. This demonstrates that the effect of free convection in the gravitational field cannot be neglected. In fact, if the molecular diffusion and the convection occurred within the boundary layer, the convection not only ran across the boundary layer, but also had a component along the layer. .
.
.
ry layers .
.~ .
lAl2~: (a) free convection; (b) forced con~ection.
3.2.2. Concentration distribution within the bounda-
l
•
/
=
. ‘~
.‘~
(~%) .
Fig. 3. The relation between the thickness of boundary layer and the supersaturation. The location on the prismatic face is 2 mm from [010].
In experiments,
the recorded interference
fringes exhibit the phase variation of the light. This variation is caused by the changes of the solution concentration within the boundary layers. The locus of the interference fringes is determined .
.
Yu Xiling et a!. / Studies ofsolute boundary layers around crystals
by the changes of the refractive index in the solution. The relation is 2K + 1 = 2d (2)
L~fl
where /.~nis the difference of refractive index, X is the wavelength of the light and d is the thickness of the phase object. When the experimental temperature is 29°C, the relation of the refractive index of the KDP solution with respect to its concentration is [9] n
=
1 3304
0 00113C
+
(3)
Carrying out the measurement of the distance of a fringe to the crystal surface, we can calculate the concentration in the position of the i th fringe. The relation is C,
=
C
—
—
2d
i~ ~
~ fl
(4) T
For example, the curve of the concentration distribution for the pyramidal face, 5 mm distance to [010] in fig. la, is shown in fig. 4a. The concentration distribution curve in the dissolving state is shown in fig. 4b. As can be seen in fig. 4, the curves of concentration distribution are nonlinear no matter whether it is in growing state or in the dissolving state. In free convection, the concentration gradients decrease slowly to zero at the pe-
riphery of the boundary layer, especially in the case of low supersatuation. This is in agreement with the result measured by the polarized filter interference method [7], in contrast to the result reported in ref. [8]. Ref. [8] used the schlieren method that cannot be used for quantitative studies, so the result obtained tends to lead to a wrong conclusion. 3.2.3. Influence offorced-convection on the boundary layer
Comparing free convection with forced convection (see figs. 2a and 2b), the 8~of the prismatic face is thinned rapidly in the vicinity of zero. When the solution was forced to convection (50 rp/min), at the same supersatuation of 1.02% the 8~of the pyramidal face decreases from 99.5 to 42.5 ~tm. The concentration, velocity and temperature for the solute are coupled to one another in the boundary layer and result in a nonlinear function.
4. Conclusion (1) Holographic phase-contrast interferometric microphotography is a technique combining phase-contrast micrography with holographic real-time interferometry. Not only can it reveal
22.5
23
__
21.?~
693
°__
100
200
300
400
distance from crystal surface x(pm)
500
0
100
200
300
400
500
distance from crystal surface x(~m)
Fig. 4. Curves of the concentration distribution within the solute boundary layer of KDP under free convection: (a) in crystal growth process; (b) in dissolving state.
694
Yu Xiling ci al.
/ Studies ofsolute boundary layers around crystals
clearly the crystal surface morphology and the characteristics of the solute boundary layers during the crystal growth processes, but it may also be used to carry out real-time measurements in various complex states for the movement. In addition, the application scope of this technique is much wider than that of classical interferometry. It is easier to be used in quantitative analysis than holographic phase-contrast microphotography, and the precision is much higher than that of holographic interferometry, due to the fact that it is visualized and convenient. (2) The experiments demonstrated that the thickness of the boundary layer increased linearly with increasing supersaturation under free convection, while the concentration distribution in the boundary layer exhibited nonlinear variations.
Acknowledgements The authors gratefully acknowledge the helpful discussions with Dr. Ren Hongwen and Professor
Jiang Huizhu on the experiment and the manuscript.
References [1] E.N. Gusava, B.M. Genzburk and B A. Kramarenko, Rost Kristallov 11(1975) 216. [2] V.A. Petrovsky, V.1. Rakin and V.P. Ruzov, J. Crystal Growth 56 (1982) 7. [3] L.J. Kuijvenhoven and l.J. Risseeuw. Intern. Sugar J. 85 (1983) 35. [4] F Bedarida, P. Boccacci and L Zefiro, Physicochem. Hydrodyn. 6 (1985) 329. [5] Yu Xiling and Wei Aijian, J. Crystal Growth 83 (1987) 11. [6] Dong Jinchang and Yu Xiling, J. Synthetic Crystals 17 (1988) 123. [7] S. Goldsztauh. R. Itti and F. Mussard. J. Crystal Growth 6 (1970) 130. [8] W.J.P. van Enckevort and M. Matuchova. Crystal Res. Technol. 22 (1987) 167. [9] Yu Xiling, Shao Zongshu and Wang Ruihua. Chem. J. Chinese Univ. 9 (1988) 1252.
Journal of Crystal Growth 106 (1990) 690—694 North-Holland
QUANTITATIVE STUDIES OF SOLUTE BOUNDARY LAYERS AROUND CRYSTALS BY HOLOGRAPHIC PHASE-CONTRAST INTERFEROMETRIC MJCROPHOTOGRAPHY
*
YU Xiling, YUE Xuefeng, GAO Hangjun and CHEN Huanchu institute of Crystal Materials, Shandong University. Jinan 250100, People’s Rep. of China Received 24 March 1989: manuscript received in final form 2 June 1990
This paper discusses an effective real-time measuring technique — holographic phase-contrast interferometric microphotography — for the studies of solute boundary layers around a crystal and the essential conditions for precise measurements under complicated conditions. The experiments demonstrated that the thickness of the solute boundary layer increased linearly with increasing supersaturation in the free-convection State while the solute distribution within the boundar~ layer exhibited nonlinear variations.
1. Introduction
2. Holographic phase-contrast interferometric microphotography
During solution crystal growth processes, at the place where solid and liquid meet, the true behaviour of solute transport remains a subject of research. In recent years, some papers reported studies on the distribution of concentration field and measurements of the thickness of boundary layers by holographic interferometry [1—4].Unfortunately, the analyses reported in these papers were performed under free-convection conditions, and they presented neither the solute distribution within the boundary layer which is an important region for solute transport, nor the regularities of variations in the boundary layers under forced convection. In addition, the accuracy of the measuring methods was not discussed in these papers. This paper presents an effective real-time measuring technique holographic phase-contrast interferometric microphotography for the measurement of solute boundary layers and reports the essential conditions needed for carrying out precise measurements. Using this technique, the authors have measured the thickness of the boundary layers and the concentration distributions within the layers under different convection conditions. —
—
*
This research project is supported by the National Nature Science Foundation of China (NSFC).
0022-0248/90/$03.50 © ~990
—
2.1. Fundamental principle
The principle of holographic phase-contrast microphotography for crystal growth research has been discribed in a previous paper [5]. Combining this principle with that of holographic interferometry, the holographic phase-contrast interferometnc microphotography can be formed, to carry out a comparison of the filtered object waves at two different time. The crystal growth causes the phase variations of the light. When the phase difference satisfies the following formula, (2K+ 1)~, (I) ~
where IK~ O~ 1, 2 interference dark fringes are formed. Placing the interference patterns under the microscope, we can observe and calculate the particular details concerned. =
2.2. The essential conditions for carrying out precise measurements on solute boundary layers
In order to study the environmental phase of a special interface, this environmental phase has to be exposed fully. The simple and reliable way is to make the crystal face parallel to the optical axis of the subject beam.
Elsevier Science Publishers B.V. (North-Holland)
Yu Xiiing et ai.
/
Studies ofsolute boundary layers around crystals
Furthermore, when the incident light radiates upon the edge of any object, the diffraction fringes will appear. These diffraction fringes can be seen clearly in the phase-contrast microscope. Only when the measuring screen is placed on the image plane, are the number and width of the diffraction fringes of the crystal edge the least and the smallest, and is at the same time, the clarity of the morphological image of the crystal optimal. Therefore, the measuring screen can be precisely adjusted and located by using the edge diffraction in the phase-contrast micrography as a criterion, In the meantime, this technique can also be useful in calibrating whether the crystal face to be measured is parallel to the optical axis. Finally, if the size of the crystal is large or the supersaturation of the growing solution is high, attention must be paid to the influence of the refraction of the light and the edge effect. It is supposed in the interference equation that the light travels straightly; in fact, when the light passes through a region where the gradient of refractive index is high, the deviation caused by the refraction cannot be neglected. On the other hand, the light passes through the two ends of the boundary layer and this is a non-two-dimensional problem, so that corrections of the edge deviations
691
should be made in order to obtain a precise conelusion [6].
3. The measurements on solute boundary layer 3.1. Experiment
The light source used is a He—Ne laser. The temperature in the crystallizer is maintained at an accuracy of ±0.05°C by a super-thermostat. The volume of the solution is 250 ml. The solution was forced to convect by the electromagnetic agitator. The crystallite seed is KDP with a size of about 6 X 6 x 6 mm. A single-exposure hologram was made to the saturated solution and processed in situ; then a crystallite seed was put into the solution, carefully adjusted and located with the aid of phase-contrast micrography. Through the changing of ternperatures, different holograms of different boundary layer variations can be obtained (see fig. 1). If the convection state of solution changed under the same supersaturation, holograms of the changed boundary layer under free convection and forced convection will be obtained (see fig. 2). The variation phenomena shown above were re-
—
—
Fig. I. Holograms of different boundary layers for different supersaturations of KDP solution: (a) in crystal growth process = 1.04%; (b) in dissolving state ~C = —0.19%. Magnification 12 X.
692
Yu Xiling et aL
/ Studies ofsolute
boundary laveri around crystals
Fig. 2. Holograms of different convection states at a temperature of 29.05°c. .~( Magnification 12 x.
corded by microphotography. After processing the films, measurements and calculations were made, 3.2. Results and discussion 3.2.1. Thickness of solute boundary layer
The thickness ~ of the solute boundary layer changes with the variations of the solution supersatuation under free convection. The experimental results are shown in fig. 3, from which we can see that 8~increases linearly with an increase of the supersatuation. This experimental result is funda-
4~
/
/
/
300
/
250
200
/
//
mentally in agreement with the diffusion equation of Nernst. The thickness 8 of the Nernst diffusion layer is in proportion to the supersatuation directly. From this point of view, the solute boundary layer is the diffusion layer. However, the experiment also shows that the thickness of the boundary layer is not equal on the different faces of a crystal and at different positions of a crystal face. As shown in fig. 2a, the thickness 8~of the boundary layer at a pyramidal face of 5 mm distance to [010] is 99.3 ~tm, and 6~at a prismatic face of the same distance to [010] is 230.9 p~m.However, the 8~at the prismatic face of 10 mm distance to [010] is 286.8 !.Lm. This is not in agreement with the Nernst diffusion equation in which 8 should be equal in value at the same prismatic face of equal area. This demonstrates that the effect of free convection in the gravitational field cannot be neglected. In fact, if the molecular diffusion and the convection occurred within the boundary layer, the convection not only ran across the boundary layer, but also had a component along the layer. .
.
.
ry layers .
.~ .
lAl2~: (a) free convection; (b) forced con~ection.
3.2.2. Concentration distribution within the bounda-
l
•
/
=
. ‘~
.‘~
(~%) .
Fig. 3. The relation between the thickness of boundary layer and the supersaturation. The location on the prismatic face is 2 mm from [010].
In experiments,
the recorded interference
fringes exhibit the phase variation of the light. This variation is caused by the changes of the solution concentration within the boundary layers. The locus of the interference fringes is determined .
.
Yu Xiling et a!. / Studies ofsolute boundary layers around crystals
by the changes of the refractive index in the solution. The relation is 2K + 1 = 2d (2)
L~fl
where /.~nis the difference of refractive index, X is the wavelength of the light and d is the thickness of the phase object. When the experimental temperature is 29°C, the relation of the refractive index of the KDP solution with respect to its concentration is [9] n
=
1 3304
0 00113C
+
(3)
Carrying out the measurement of the distance of a fringe to the crystal surface, we can calculate the concentration in the position of the i th fringe. The relation is C,
=
C
—
—
2d
i~ ~
~ fl
(4) T
For example, the curve of the concentration distribution for the pyramidal face, 5 mm distance to [010] in fig. la, is shown in fig. 4a. The concentration distribution curve in the dissolving state is shown in fig. 4b. As can be seen in fig. 4, the curves of concentration distribution are nonlinear no matter whether it is in growing state or in the dissolving state. In free convection, the concentration gradients decrease slowly to zero at the pe-
riphery of the boundary layer, especially in the case of low supersatuation. This is in agreement with the result measured by the polarized filter interference method [7], in contrast to the result reported in ref. [8]. Ref. [8] used the schlieren method that cannot be used for quantitative studies, so the result obtained tends to lead to a wrong conclusion. 3.2.3. Influence offorced-convection on the boundary layer
Comparing free convection with forced convection (see figs. 2a and 2b), the 8~of the prismatic face is thinned rapidly in the vicinity of zero. When the solution was forced to convection (50 rp/min), at the same supersatuation of 1.02% the 8~of the pyramidal face decreases from 99.5 to 42.5 ~tm. The concentration, velocity and temperature for the solute are coupled to one another in the boundary layer and result in a nonlinear function.
4. Conclusion (1) Holographic phase-contrast interferometric microphotography is a technique combining phase-contrast micrography with holographic real-time interferometry. Not only can it reveal
22.5
23
__
21.?~
693
°__
100
200
300
400
distance from crystal surface x(pm)
500
0
100
200
300
400
500
distance from crystal surface x(~m)
Fig. 4. Curves of the concentration distribution within the solute boundary layer of KDP under free convection: (a) in crystal growth process; (b) in dissolving state.
694
Yu Xiling ci al.
/ Studies ofsolute boundary layers around crystals
clearly the crystal surface morphology and the characteristics of the solute boundary layers during the crystal growth processes, but it may also be used to carry out real-time measurements in various complex states for the movement. In addition, the application scope of this technique is much wider than that of classical interferometry. It is easier to be used in quantitative analysis than holographic phase-contrast microphotography, and the precision is much higher than that of holographic interferometry, due to the fact that it is visualized and convenient. (2) The experiments demonstrated that the thickness of the boundary layer increased linearly with increasing supersaturation under free convection, while the concentration distribution in the boundary layer exhibited nonlinear variations.
Acknowledgements The authors gratefully acknowledge the helpful discussions with Dr. Ren Hongwen and Professor
Jiang Huizhu on the experiment and the manuscript.
References [1] E.N. Gusava, B.M. Genzburk and B A. Kramarenko, Rost Kristallov 11(1975) 216. [2] V.A. Petrovsky, V.1. Rakin and V.P. Ruzov, J. Crystal Growth 56 (1982) 7. [3] L.J. Kuijvenhoven and l.J. Risseeuw. Intern. Sugar J. 85 (1983) 35. [4] F Bedarida, P. Boccacci and L Zefiro, Physicochem. Hydrodyn. 6 (1985) 329. [5] Yu Xiling and Wei Aijian, J. Crystal Growth 83 (1987) 11. [6] Dong Jinchang and Yu Xiling, J. Synthetic Crystals 17 (1988) 123. [7] S. Goldsztauh. R. Itti and F. Mussard. J. Crystal Growth 6 (1970) 130. [8] W.J.P. van Enckevort and M. Matuchova. Crystal Res. Technol. 22 (1987) 167. [9] Yu Xiling, Shao Zongshu and Wang Ruihua. Chem. J. Chinese Univ. 9 (1988) 1252.