Optical measurement techniques based on changesin the refractive index of the medium find utility in avariety of engineering applications. One area is on-line monitoring of the growth process of a transpar-ent crystal from its aqueous solution. Such crystalswith a high degree of perfection are required in ap-plications ranging from proteins to high-power la-sers. A crystal growing from its aqueous solutioncreates a three-dimensional solute distribution in itsvicinity. The solutal concentration gradients, andhence gradients in the density of the solution, areresponsible for the evolution of buoyancy-driven con-vection currents in the growth chamber. Mapping ofconvection patterns as well as the determination ofthe three-dimensional concentration fields in the
growth chamber are required to understand themechanism of crystal growth and to establish appro-priate conditions for growing large defect-free crys-tals.
A large amount of recent literature is available onthe application of refractive-index-based methods tothe study of three-dimensional buoyancy-driven con-vective fields. Snyder and Hesselink1 carried out in-stantaneous optical tomographic reconstruction ofnonstationary fluid flow. The authors determined thespatial distribution of helium concentration in a co-flowing jet of helium in air. Holographic interferom-etry was used to measure the optical path lengthalong several views. Gatti et al.2 carried out three-dimensional measurements of concentration fields incrystal growth using multidirectional holographic in-terferometry. The authors employed the sinc methoddeveloped by Sweeney and Vest3 as the reconstruc-tion algorithm for discrete data collected over a lim-ited range of view angles. Watt and Vest4 studiedstructures of turbulent helium jets in air by recordingpath integral images based on the refractive-indexvariation with a pulsed phase-shifted interferometer.Subsequently, the authors reconstructed the heliumconcentration field over the jet cross section. Michael
The authors are with the Department of Mechanical Engineer-ing, Indian Institute of Technology Kanpur, Kanpur 208016, India.The e-mail address for K. Muralidhar is [email protected].
Received 7 January 2005; revised manuscript received 15 April2005; accepted 20 April 2005.
and Yang5 discussed measurement of the three-dimensional temperature field in Rayleigh–Benardconvection. A Mach–Zehnder interferometer wasused by the authors, and reconstruction was per-formed using an iterative technique. McMackin andHugo6 employed a high-speed optical tomographysystem for imaging dynamic transparent media. Thesystem was used to obtain spatially resolved high-speed cross-sectional temperature images of a roundheated jet. Agrawal et al.7 employed quantitativerainbow schlieren deflectometry with tomography formeasurements of temperature in three-dimensionalgas flows. The beam deflection data acquired fromschlieren images at multiple viewing angles wereused to reconstruct refractive-index and temperaturedistributions with a modified convolution backprojec-tion (CBP) tomographic algorithm. Ko and Kihm8
applied laser speckle photographic tomography to re-construct the concentration field generated by asym-metric single and double helium jets in air. Mishra etal.9 conducted an interferometric study of Rayleigh–Benard convection in air at intermediate Rayleighnumbers using tomography with limited projectiondata. Notcovich et al.10 investigated the three-dimensional temperature field around heavy ice�D2O� growing from supercooled heavy water usinginterferometric tomography. The authors illustrateduse of a temperature map to understand the stabilityof asymmetrical morphologies that have been ob-served in ice and other crystals. Dezhong andTiange11 applied real-time laser interferometric to-mography for the measurement of three-dimensionaltemperature fields generated by two heated copperrods. More recently, the problem of handling incom-plete and limited projection data has been addressedby Mishra et al.12 wherein the authors carried out adetailed evaluation of iterative tomography algo-rithms for convection in a cylindrical cavity.
In the context of crystal growth, Onuma et al.13 car-ried out a microscopic study of barium nitrate andK-alum crystals using schlieren and Mach–Zehnderinterferometry. The authors studied the effect ofbuoyancy-driven convection and forced flow rate on themicrotopography of the crystal growing from its aque-ous solution. Masayuki et al.14 employed a Mach–Zehnder interferometer for the measurement ofconcentration gradients around a potassium dihydro-gen phosphate (KDP) crystal growing from its aqueoussolution. Kleine et al.15 utilized a dark field-typeschlieren microscope for quantitative imaging of soluteconcentration profiles around a growing crystal. Inoueet al.16 employed the schlieren technique along with aMach–Zehnder interferometer for in situ investigationof spherulites during the growth of Na2S2O3.5H2O.Srivastava et al.17 compared schlieren, shadowgraph,and interferometry techniques for on-line monitoringof the growth process of a KDP crystal from its aque-ous solution. On the basis of the ease of instrumen-tation, image clarity, and simplicity of data analysis,the authors suggested the laser schlieren techniqueas the most suited for the monitoring of the growthprocess. In a subsequent study, the authors mapped
the concentration field around stationary and rotat-ing crystals growing from their aqueous solution.18
Unlike growth from melt and vapor, growth of acrystal from its aqueous solution is particularly ame-nable to optical visualization, since the solution istransparent. It is possible to generate images of theconvective field by exploiting changes in the refractiveindex that accompany changes in the density of themedium.19 Recorded images can be related to the pathintegral of the field variable or its derivatives. Thisinterpretation permits the analysis of the full three-dimensional field of the variable being measuredthrough use of various tomographic reconstructionalgorithms.20–22 The present work employs a CBP al-gorithm for the reconstruction of a three-dimensionalconcentration field in the growth chamber. Herman20
has discussed the convergence properties of the algo-rithm. Error analysis in terms of the filter propertieshas been reported by Munshi.23,24
In the present study, the laser schlieren techniqueis employed to investigate the three-dimensional con-centration field around a KDP crystal growing fromits aqueous solution. The choice of the KDP crystal forexperiments was dictated by the availability ofrefractive-index data for the solution at various tem-peratures and concentrations. The convective field isset up in the growth chamber by inserting a KDPseed into its supersaturated solution followed by slowcooling of the solution. The projection data in theform of two-dimensional schlieren images are re-corded from four different view angles (0°, 45°, 90°,and 135°) by turning the crystal growth chamber. Theintegrated values of concentration are obtained byanalyzing the light intensity data of the schlierenimages. Subsequently, the concentration fields atvarious horizontal planes above the crystal are recon-structed using the CBP algorithm. Because of thelimitations in the optical system, the projection datarequired for tomographic inversion are often incom-plete. The applicability of the inversion algorithmscan then be ensured only when additional tests areconducted to validate the result obtained. The vali-dation with numerically simulated as well as exper-imental data is reported in this work. The focus of thepresent study is to examine the symmetry of the con-centration field in the vicinity of the growing crystalin the initial stages (diffusion-dominated growth) andin the stable growth regime (recognized by the pres-ence of a steady convective plume rising from thecrystal surface). Of interest is the relationship be-tween the morphology of the crystal and the solutalconcentration field around it during the growth pro-cess.
2. Apparatus and Instrumentation
The crystal growth chamber used in the present ex-periment is shown schematically in Fig. 1. It com-prises a glass chamber that holds the KDP solutionand has a diameter of 16.5 cm with a height of 23 cm.For visualization of the concentration field by theschlieren technique, circular optical windows (BK-7,40 mm in diameter, 5 mm thick, ��4) are fixed on the
glass beaker at opposite ends. A total of eight suchwindows permit passage of the laser beam for fourview angles. Parallelism and straightness of the op-tical windows are crucial for the generation of mean-ingful images, and considerable precautions havebeen taken in this regard. The Plexiglas tank sur-rounding the growth chamber is octagonal in plan. Itensures a large enough volume for the circulatingthermostated water to keep the KDP solution at therequired temperature level over a period of time.Four heating elements placed diametrically oppositein the outer chamber maintain the temperature ofcirculating water, and hence the KDP solution. Elec-trical input to the heating elements is regulated by aprogrammable temperature controller (Eurotherm).A K-type thermocouple wire fixed to the outer surfaceof the growth chamber provides the feedback to thecontroller. Uniformity of temperature within the so-lution is ascertained by recording temperatures atvarious locations using 26-gauge K-type thermocou-ples. With this arrangement, it was possible to reducethe temperature of the solution from 36 °C to 25 °Clinearly with time over a period of 60 h.
The salt solution has an initial temperature that ishigh enough to keep it from becoming fully saturated.A KDP seed crystal spontaneously crystallized in asecond vessel is placed on a glass platform and intro-
duced in the growth cell. When the bulk temperatureis lowered, the solution becomes supersaturated withsalt that in turn deposits on the crystal. A crystalgrowing from its aqueous solution thus creates athree-dimensional solute distribution in its vicinity.The solutal concentration gradients, and hence thegradients in the density of the solution, are respon-sible for the evolution of buoyancy-driven convectioncurrents in the growth chamber. The buoyant con-vection currents influence the magnitude of the con-centration gradients prevailing along the growth in-terfaces. In turn, these control the stability of thegrowth process and the overall crystal quality.
The seed crystal is a square with a 3 mm edge andis 1 mm thick. With deposition taking place over it,the crystal becomes pyramidal in shape and retainsthe shape for the entire duration of the experiment.The present study is concerned with the convectivefield away from the crystal, and crystal morphology isnot of direct interest.
For optical measurement of the convective field,a continuous-wave helium–neon laser (Spectra-Physics, 35 mW) was employed as the coherent lightsource. A monochrome CCD camera (Sony) with aspatial resolution of 768 � 574 pixels was used torecord the convective field in the form of two-dimensional images. One such image is recorded foreach pair of optical windows. The time lapse involvedin turning the apparatus is not important becausethe crystal growth process is quite slow. The camerawas interfaced with a personal computer (HCL, 256MB RAM, 866 MHz) through an 8 bit frame grabbercard. Light intensity levels were digitized over therange of 0–255. Image acquisition was at video ratesof 25 frames/s. Schlieren images displayed in thiswork have been subjected to image processing oper-ations to improve contrast; the data analysis for re-covering the field concentration, however, is based onthe intensity changes in the original images.
3. Optical Arrangement
A monochrome schlieren technique has been used inthe present work to study convective phenomena inthe crystal growth process. Figure 1 includes the lay-out of a Z-type laser schlieren setup after integrationwith the crystal growth cell. The optical componentsinclude concave mirrors of 1.30 m in focal length and200 mm in diameter. Relatively large focal lengthsmake the schlieren technique sensitive to the concen-tration gradients.19 The knife edge is placed at thefocal length of the second concave mirror. It is posi-tioned to cut off a part of the light focused on it sothat, in the absence of any optical disturbance, theillumination on the screen is uniformly reduced. Theinitial intensity values in the experiment are chosento be less than 20 on a gray scale of 0–255. To avoidcamera saturation, experiments are conducted insuch a way that the maximum light intensity is lessthan 200. In some experiments, a neutral-density fil-ter has been employed for this purpose. The knifeedge is oriented perpendicular to the direction inwhich the density gradients in the aqueous solution
Fig. 1. Schematic diagram of the four-view crystal growth cham-ber placed in the path of the laser beam in a Z-type schlieren setup.(1) Growing crystal on a platform, (2) growth chamber, (3) outerchamber, (4) heating element, (5) thermocouple, (6) optical win-dow, (7) seed holder (platform configuration), (8) covering lid, (9)temperature controller unit.
are to be recorded. In addition, the field of view in-cludes the undisturbed solution away from the crys-tal, where the solutal concentration is a knownquantity. In the present study, the gradients are ex-pected to be predominantly in the vertical directionparallel to the gravity vector, and the knife edge hasbeen kept horizontal.
4. Data Reduction
The images recorded by the schlieren apparatus yieldthe line-integrated values of the concentration gradi-ent. The integration referred to here is in the direc-tion of propagation of the light beam, and gradientsare in the vertical direction. Data analysis refers torecovering the line-integrated concentration values,followed by the calculation of local concentration dis-tribution using principles of tomography.
Image formation in a schlieren system is due to thedeflection of the light beam in a variable refractive-index field toward the region of higher refractive in-dex. To perform a quantitative analysis of a schlierenimage, the cumulative angle of refraction of the lightbeam emerging from the growth chamber has to bedetermined as a function of position, in the planenormal to the light beam. The direction of propaga-tion of light is indicated as z, and the concentrationdistribution is obtained in the x–y plane. The result ofthe analysis is a concentration field that is ray aver-aged in the z direction over the diameter of thegrowth chamber.
Using principles of ray optics, the total angulardeflection � of a light ray during its passage withinthe aqueous solution (but excluding the optical win-dow) can be expressed as19,25
� �1na�
0
L
n�(ln n)
�y dz.
The intensity field can now be related to therefractive-index field directly as
�IIk
�f
akna�
0
L�n�y dz. (1)
Here na is the refractive index of the environment andis practically unity. Furthermore, ak is the verticalsize of the focal spot after being partly blocked by theknife edge and f is the focal length of the decollimat-ing mirror. The integral expression can be resolved byexpressing Eq. (1) in terms of the gradient of theray-averaged refractive index. The resulting equationfor the schlieren process can be derived as
�IIk
�f
ak
�n�y L. (2)
Here L is the length of the path traversed by the lightbeam through the growth chamber, namely, its di-
ameter. Equation (2) requires the approximation thatchanges in the light intensity occur due to beam de-flection, rather than its physical displacement.
The concentration gradient is related to therefractive-index gradient of the KDP solution usingthe following relation14:
�N�y �
9n
2�(n2 � 2)2
�n�y . (3)
Here N is the molar concentration of the solutionand � is the polarizability of the KDP crystal�� 4.0 cm3�mol�. Combining Eqs. (2) and (3) and in-tegrating from an outer location in the bulk of thesolution (where the gradients are negligible), the con-centration distribution around the growing crystalcan be uniquely determined. The contribution of re-fraction of light at the confining optical windows(BK-7) has been accounted for by applying a correc-tion factor in Eq. (2) for the angle of deflection withwhich the beam emerges from the growth chamber. Itcan be shown using Snell’s law that the correctionfactor is equal to the refractive index of the KDPsolution �� 1.355� at the ambient temperature.
The schlieren images have been utilized to recon-struct the concentration field on horizontal planesabove the crystal. For this purpose, images have beenrecorded at four different view angles. Since theseimages are time separated, the present study is re-stricted to steady convection conditions. Minorchanges in the concentration field with time havebeen accounted for by averaging a sequence of foursuccessive images. The schlieren images recordedduring the initial transients as well as late stages ofgrowth when convection is unsteady have not beenutilized for three-dimensional reconstructions.
A. Computed Tomography
The schlieren image yields gradients of refractive in-dex and hence concentration. As discussed above, thegradient information can be integrated to yield datain terms of concentration itself. Subsequently, theconcentration field on selected horizontal planesabove the growing crystal can be reconstructed usingprinciples of tomography.20 For this purpose, the leftside of Eq. (2) is interpreted as projection data, ini-tially of the refractive-index gradient, and after inte-gration, for the concentration field.
Tomography has been used in the present work toreconstruct two-dimensional concentration fieldsover individual horizontal planes from their one-dimensional projections. The recording configurationshown in Fig. 1 is parallel beam geometry. Recon-struction is sequentially applied from one plane to thenext in the third (vertical) direction. The CBP algo-rithm has been used in the present study for tomo-graphic reconstruction. Significant advantages of thismethod include its noniterative character, availabil-ity of analytical results on convergence of the solutionwith respect to the projection data, and establishederror estimates.20–24 The treatment of reconstruction
for partial data adopted in the present work differsfrom that of Mishra et al.,12 where an iterative ap-proach was followed.
In an experiment, projection data can be recordedeither by turning the crystal growth chamber or thesource–detector combination. The latter is particu-larly difficult because of the stringent requirement ofalignment. With the first option, it is not possible torecord a large number of projections, owing to theinconvenience of installing plane optical windows in acircular beaker. Furthermore, for a cylindricalgrowth chamber, the entire field of interest cannot beimaged due to the curvature of the test cell. Instead,the central core region (corresponding to the size ofthe optical windows), which includes the growingcrystal, has been recorded. In this respect, the pro-jection data set is incomplete. To generate a completeprojection data set for each view angle, an extrapo-lation procedure has been adopted (Subsection 4.B).
Projection data were recorded for four view anglesof 0°, 45°, 90°, and 135° in the present experiments.The data for 180° is taken to be identical to that forzero angle. Information for intermediate view angleshas been generated by employing linear interpolationon the experimentally recorded data.
The experimental setup employed in the presentwork allows only a part of the aqueous solution in thebeaker to be scanned by the laser beam, being limitedby the size of the optical windows. The projection dataare thus incomplete, as shown schematically in Fig.2. To successfully apply the CBP algorithm for tomo-graphic inversion, one needs projection data over theentire width of the physical domain for each viewangle. In the present work, the experimentally re-corded partial data have been extrapolated to deriveinformation about the portion of the solution beyondthe optical windows. In all the experiments, the re-gion inclusive of the crystal was imaged; extrapola-tion is applied to the portion of the field from theedges of the schlieren image to the walls of the bea-ker.
The applicability of extrapolation to the presentstudy can be justified on the basis of two factors:
First, the concentration level away from the growingcrystal, corresponding to the supersaturated solu-tion, is practically constant. This is confirmed in therecorded schlieren images, where the changes in theintensity are found to be localized in the vicinity ofthe growing crystal alone. These images are dis-cussed in the subsequent sections. Second, the infor-mation content of schlieren images decreases withthe geometric path length of the light beam withinthe beaker. This is indicated by the integration limitsof Eq. (1). The diminishing chord length of the beakertoward the edges shows that the measurement pro-cedure de-emphasizes concentration variation occur-ring toward the sides of the beaker.
In the present work, a tenth-order polynomial hasbeen used to extrapolate the concentration distribu-tion within the region covered by the windows. Thelimiting values of concentration in the far field, andthe necessity of maintaining slope continuity in theconcentration distribution at every point, have beenenforced. Mass balance errors in solutal concentra-tion were better than 0.01% in all the experimentsdiscussed in the present work.
B. Convolution Backprojection Algorithm
In the CBP algorithm, the reconstructed functionf�r, � (namely, refractive index for the present study)is evaluated by the integral formula20,23,24:
f(r, ) ��0
��D�2
D�2
p[(s� � s);�]q(s)dsd�, (4)
where
q(s) ��Rc
Rc
|R|W(R)exp(i2Rs)dR.
Here p�s; �� is the projection data of the refractive-index field within the solution and s is the perpen-dicular distance of the data ray from the center of theobject. Furthermore, � denotes the source positionwith respect to a fixed axis, D is the diameter of thegrowth chamber, s� is the s value of the data raypassing through the point �r, �. The Fourier fre-quency is denoted by R, q�s� is the convolving func-tion of Eq. (4), and W�R� is the filter function. Thefilter function vanishes outside the interval ��Rc,�Rc� and is an even function of R. Here Rc is theFourier cutoff frequency and is taken to be 1�2�s, �sbeing the ray spacing. The reconstruction obtained isspecific to the choice of the filter function.20 A Ham-ming filter h54 was used in the present study, and,for |R| Rc, it is given by the formula
W(R) � 0.54 � (1 � 0.54)cos�RRc
�.As discussed by Natterer21 and Munshi,23,24 this filterFig. 2. Definition of partial projection data.
emphasizes the smoother aspects of the concentra-tion variation while suppressing small-scale (fine-structure) fluctuations. This is quite appropriate tothe present study for the following reason. Densityvariations in the solution arise primarily because ofthe deposition of the solute on the growing faces of thecrystal as the solution is cooled at a given rate. Sincethe crystal growth rate is slow, one encounters den-sity variations that are distributed in the entire so-lution, but rapid fluctuations in concentration do notappear.17,18 Hence it is of interest to reconstruct thedominant pattern in the concentration field ratherthan its secondary features.
C. Validation of the Reconstruction Procedure withSimulated Data
The goal of the present work is to obtain a concen-tration distribution on selected planes above the crys-tal growing from its aqueous solution from theschlieren images. These images must be extrapolatedto fill the width of the beaker. The extrapolation stepcombined with the CBP algorithm is first validatedagainst simulated data. The physical problem consid-ered is buoyancy-driven convection in a differentiallyheated circular fluid layer with upper and lower wallsmaintained at specified temperatures. The sidewallis thermally insulated. The fluid considered is air andthe Rayleigh number based on the height of the fluidlayer is set at Ra � 12,000. The temperature distri-bution in the fluid layer has been obtained by numer-ically solving the governing equations of flow andenergy transport on a fine grid. For definiteness, thethermal field is taken to be axisymmetric; accordinglythe isotherms on individual planes of the fluid layerare circular. The physical realizability of axisymmet-ric convection in a circular cavity around a Rayleighnumber of 12,000 has been discussed at length byVelarde and Normand26 and Leong.27
With the solution for temperature determined nu-merically, the projection of the thermal field is ob-tained by path integration. The accuracy ofreconstruction with partial data is examined in thepresent study against the available numerical solu-tion. Errors are reported in the present section onthree grids, namely, 64 � 64, 128 � 128, and 256� 256. Here the first number represents the numberof view angles along which projections have been re-corded, and the second indicates the number of raysfor each view. The definitions of errors considered are
Here Torig and Trecon are the original and reconstructedtemperature fields, respectively, and N is the totalnumber of grid points on the reconstructed plane. Alltemperatures generated by simulation are dimen-
sionless and in the range of 0 to unity. The differencebetween error norms E1 and E2 arises from the factthat the former highlights large isolated errors,whereas the latter reveals trends that are applicablefor the entire cross section.
The validation of the reconstruction procedure withsimulated data is summarized in Fig. 3. Figure 3(a)shows isotherms of the thermal field in the fluid layerfor a given view angle. These data are presented inthe form of contours of the path-integrated tempera-ture field. Since the thermal field is axisymmetric,the projection data for all other view angles are iden-tical to Fig. 3(a). The reconstruction over a horizontalplane of the fluid layer is shown in Fig. 3(b). Resultsobtained with complete projection data [100% in Fig.3(b)(i)] and partial data [60% in Fig. 3(b)(ii) and 30%in Fig. 3(b)(iii), symmetrically placed about the cen-ter] are also shown. The axisymmetric nature of tem-
Fig. 3. Buoyancy-driven convection in a differentially heated cir-cular cavity. (a) Complete projection data in the form of isothermsfor the differentially heated circular fluid layer; (b) reconstructedtemperature contours at y�H � 0.65 for full (i) 100% and partial[(ii) 60%, (iii) 30%] projection data; (c) comparison of original andreconstructed nondimensional temperature distribution along theradial direction for the three different combinations of rays andviews.
perature distribution is brought out in all thereconstructions. This can be taken as a validation ofthe extrapolation procedure used to convert a partialto an approximate but complete data set. A quanti-tative comparison of the reconstructed temperatureprofiles along the diameter of the cavity for differentcombinations of rays and views is shown in Fig. 3(c).For the complete data set, a perfect match betweenthe original and the reconstructed profiles can beseen for grid sizes of 128 � 128 and 256 � 256,whereas small errors are seen for the 64 � 64 grid.The extent of deviation from the original increases asthe fraction of incomplete data increases. Noticeableerrors are to be seen when only 30% of the originaldata is used, the rest of it being derived by extrapo-lation. Errors in reconstruction were found to be sig-nificantly higher when the partial data set was usedwithout extrapolation.
The magnitudes of errors as a function of discreti-zation of the fluid layer and size of the partial data setare summarized in Table 1. Since the difference be-tween the minimum and the maximum temperaturesis unity, the percentage error is obtained as 100� E1 and 100 � E2. In Table 1, error E1 is consistentlyseen to be higher than E2, the latter being an averageover the entire field. Both errors decrease as the gridsize (number of rays and views) increases. For a givengrid size, errors increase as the fraction of the origi-nal data used in reconstruction decreases. When only30% of the original is used (the rest being obtained byextrapolation), the maximum errors on a 64 � 64 gridare 17.3% (absolute maximum) and 7.6% (rms). Fig-ure 3 shows that the corresponding reconstructionsare qualitatively meaningful, and hence these errormagnitudes may be taken to be within limits.
5. Results and Discussion
Results are presented in this section for a validationexperiment, followed by detailed experiments oncrystal growth in the steady convection regime. InSubsection 5.A the validation experiment discussedis comprised of recording schlieren images as projec-tion data by turning the crystal through a sequence ofangles. Here the crystal growth chamber is keptfixed, and only one pair of optical windows is utilized.
Twelve such projections could be recorded with thisapproach. Reconstructions with two subsets of tenprojections each are compared to validate the overallapproach to measurement of the concentration field.In Subsection 5.B results are presented in the form ofa time sequence of schlieren images recorded at fourview angles, corresponding to four pairs of opticalwindows of the growth chamber. The crystal is notdisturbed during the growth process. In this respect,the measurement process is truly noninvasive, al-though with a drawback of allowing only a limitednumber of projections. The emphasis is toward study-ing the stable convection regime of the growth pro-cess. The reconstruction of the concentration fieldover selected planes above the crystal and their rela-tionship to the crystal geometry are discussed.
A. Validation Experiment with Crystal Growth
In this subsection we discuss reconstruction of theconcentration field using projection data recorded byturning the growing crystal through small angles.The growth chamber was held stationary during thecourse of the experimental run time. This configura-tion permits the recording of a larger number of pro-jections of the field under study by a single source–detector combination. On the other hand, thedisadvantage of this approach is that the flow field inthe vicinity of the growing crystal is disturbed duringthe movement of the seed holder platform. Since thegrowth process of the crystal from its aqueous solu-tion is slow, the disturbance is not expected to havean effect on the overall growth pattern. It was seen inthe experiments that the unsteadiness induced bythe disturbance faded away in �10 min.
Figure 4 shows 12 schlieren images of the concen-tration gradients recorded by turning the growingcrystal from 0° to 180°. The images shown are froman experiment conducted over a duration of 2 h, afterthe insertion of a medium-sized crystal (5 mm edge)in the aqueous solution. The field could not bescanned for view angles falling in the range of70°–120° because of the interference of the platformsupport with the laser beam. The schlieren imagesrecorded from the view angles that are close to eachother show similar distribution of intensity contrastand hence concentration gradients around the grow-ing crystal. Noticeable differences are seen in thepatterns for view angles that are widely separated.The images were practically steady in time, but aweak convection plume appeared around the crystal.The characteristics of fluid motion around the crystalas a function of its size have been discussed by theauthors elsewhere.18
Figure 5 shows the reconstructed concentrationcontours at six horizontal planes above the growingcrystal. Here the y coordinate is measured from theupper surface of the crystal, and H is the image sizeas seen through the optical window. The outermostcircle corresponds to the beaker diameter. The distri-bution of concentration closer to the crystal (y�H� 0.05 and 0.15) shows three dimensionality, specif-ically a considerable deviation from axisymmetry. At
Table 1. Comparison of the Original and Reconstructed TemperatureFields in Terms of Errors E1 and E2 for Buoyancy-Driven Convection in
a Circular Cavity
Date Type Rays � Views E1 E2
Full data 256 � 256 0.052 0.028128 � 128 0.109 0.05664 � 64 0.124 0.058
distances farther away from the crystal, a structurecloser to axisymmetric, where the contours approacha circular shape, is realized. In the flooded map,bright regions correspond to a solution depleted of thesolute, and the dark regions represent supersatu-rated solution. Figure 5 shows a gradual increase indarkness with distance from the crystal face. Hencethe average solute concentration increases with the ycoordinate. This top-heavy configuration is gravita-tionally unstable and can result in buoyancy-drivenconvection within the solution.
To validate the reconstruction procedure alongwith the extrapolation discussed in Subsection 4.B,results obtained with two subsets of projection datahave been compared. The first set comprises the im-ages recorded at angles of 0°, 10°, 20°, 30°, 40°, 140°,150°, and 180°. For the second set, view angles of 30°and 40° are substituted by those recorded at 50° and130°. The reconstructions of the concentration field attwo of the selected planes (y�H � 0.05 and 0.90) arepresented in Fig. 6. These contours are qualitativelysimilar and match those of Fig. 5, where the completedata set has been utilized. A direct comparison of theconcentration profiles obtained with two subsets ofprojection data is shown in Fig. 6(c) at three planesabove the crystal. The close match between the nu-
merical values of the concentration contours vali-dates the measurement procedure, data analysis, andthe reconstruction strategy.
B. Convection around a Growing KDP Crystal
Imaging of convection patterns around the growingKDP crystal is discussed here. Four view angles areconsidered for tomographic reconstruction of the con-centration field. In the experiments discussed, thecrystal is not disturbed during the recording of theprojection data. In addition, longer durations of crys-tal growth have been considered. Since finite time isrequired to turn the beaker and record projections,experiments have been conducted when the concen-tration field around the crystal is nominally steady.Under convection conditions, the plume above thecrystal is marginally unsteady. Here a time-averagedsequence of schlieren images has been used for anal-ysis. The number of view angles being four, the in-crement in the view angles is 45°, covering the rangeof 0°–180°.
The crystal growth process has been initiated byinserting a spontaneously crystallized KDP seed intoits supersaturated solution at an average tempera-ture of 35 °C. This step is followed by slow cooling ofthe aqueous solution. The cooling rate employed is0.05 °C/h. The seed thermally equilibrates with the
Fig. 4. Schlieren images of the convective field above the growingcrystal as recorded by turning the crystal at increments of 10°�stepwhile keeping the growth chamber fixed. The projection data forthe view angles from 70° to 120° could not be recorded because ofthe interference of the seed holder glass rod with the laser beam.
Fig. 5. Reconstructed concentration fields at various horizontalplanes above the growing crystal. The projection data in the formof two-dimensional schlieren images were recorded by turning thegrowing crystal at increments of 10°�step while keeping thegrowth chamber fixed.
solution in �20 min. With the passage of time, thedensity differences within the solution are solely dueto concentration differences. Adjacent to the crystal,the solute deposits on the crystal faces, and the solu-tion goes from the supersaturated to the saturatedstate. Thus the solution near the crystal is lighterthan the solution away from it. The denser solution inthe beaker displaces the lighter solution close to thecrystal, and for normal levels of supersaturation, acirculation pattern is set up. The onset of fluid motionis determined by the relative magnitudes of the driv-ing buoyancy force and the resisting viscous force.28
The buoyant plume resulting from fluid motion isessential to transport the solute from the bulk of thesolution to the crystal and determines the crystalgrowth rate at later stages of growth. The plume isvisible as the spread of light intensity in the schlierenimages.
One parameter that defines the quality of the grow-ing crystal is the symmetry of its faces. Since growth
from an aqueous solution is governed by the strengthand movement of buoyancy-driven convection cur-rents, the convective field plays an important role intransporting solute from the bulk of the solution tothe crystal faces. Hence, to ensure symmetric growthof the crystal, it is necessary to ensure that the con-vective field in the growth chamber retains a symme-try pattern. In the present work, a viewpoint that anaxisymmetric convective field is favorable for crystalgrowth has been adopted. Thus conditions underwhich the solutal distribution is axisymmetric havebeen investigated from the tomographic reconstruc-tions.
C. Growth Patterns in the Convection Regime
Crystal growth from a solution in which the limit ofsupersaturation has been reached is considered. Thedensity differences as well as the crystal size arelarge enough to sustain fluid motion in the beaker.Experiments were conducted under conditions of asteady plume rising from the crystal. Results for 30 hof growth are presented. For times greater than 30 h,the plume became quite unsteady. Consequently, the
Fig. 6. Comparison of the reconstructed concentration field overtwo planes (y�H � 0.05 and 0.90) above the growing crystal withtwo sets of projection data. Combinations used are (a) 0°, 10°, 20°,30°, 40°, 140°, 150°, 180° and (b) 0°, 10°, 20°, 30°, 50°, 130°, 150°,180°. (c) Concentration profiles along the central sector in thehorizontal plane for the two combinations of projection data aty�H � 0.05 and y�H � 0.90.
Fig. 7. Schlieren images of the convective field around the crystalgrowing from its aqueous solution as recorded from four differentview angles at two different time instants (2 and 30 h).
projection data of individual view angles became un-correlated.
Figure 7 shows the convective field in the form ofschlieren images. A well-defined plume rising fromthe top surface of the crystal is seen at both times oft � 2 and 30 h for four view angles. At t � 2 h, theplume was temporally stable, although a swayingmovement was seen from one projection angle to theother. The swaying motion was due to the distur-bance of the flow field caused by the turning of thegrowth chamber for recording the projection data.The tilt in the plume at selected angles for small timeis also related to the nonsymmetric shape of the ini-tial crystal itself. At a later time �t � 30 h�, the gra-dients are large enough to give rise to a stable andsymmetric convective field. This is accompanied byuniform deposition of the solute on the crystal sur-faces.
Figure 8 shows concentration contours around thegrowing crystal derived from the schlieren images ofFig. 7. The physical extent of the convective plume in
the vertical direction can be estimated by the defor-mation of the concentration contours. Densely placedcontours in the crystal vicinity are due to high-concentration gradients. The contours at t � 30 h areseen to spread into the growth chamber, indicatingthat solute from the bulk of the solution itself is de-posited over the crystal. The appearance of doublepeaks in the contour shape at the end of 30 h confirmsthat the highest gradients and hence the depositionrates occur toward the edges of the crystal.
Figure 9 shows reconstructed concentration pro-files at selected horizontal planes above the growingcrystal for the two time instants considered in Fig. 7.The following observations can be recorded on thebasis of the reconstruction:
1. At t � 2 h, the reconstructed concentration fieldreveals symmetry for the plane close to the crystal�y�H � 0.05� both around the crystal and in the bulkof the solution. The presence of concentric rings indi-cates uniform distribution of concentration gradientson all the faces of the growing crystal. This feature is
Fig. 8. Concentration contours around a growing crystal in thepresence of a well-defined convective plume for the four view an-gles at two different time instants.
Fig. 9. Reconstructed concentration profiles over five horizontalplanes �y�H � 0.05–0.90� above the crystal growing in the presenceof a stable convection plume. Time instants of 2 and 30 h of ex-perimental run time are presented.
seen in Fig. 7 as well, where the bright intensityregion uniformly envelops the crystal faces. Forplanes away from the crystal �y�H � 0.30�, the de-formation of the concentric rings in the vicinity of thecrystal is mainly due to the mild swaying motion ofthe convective plume introduced by turning thegrowth chamber. This movement can also be seen inthe schlieren images of Fig. 7. On the other hand, thebulk of the solution closer to the beaker shows uni-form and symmetric distribution of concentration.The appearance of only a few contours in the solutionshows low-concentration gradients here in the initialstages of growth.
2. For all planes �t � 2 h�, the presence of a brightcentral core of the reconstructed cross section is in-dicative of the solution depleted of salt. The spatialextent of the bright region at the center of the beakeris small, in agreement with the thin vertically risingbuoyant plume above the growing crystal (Fig. 7).
3. At later times �t � 30 h�, the concentration fieldat planes near the crystal �y�H � 0.30� is not axisym-metric, although the bulk of the solution near theboundaries of the growth chamber shows the pres-ence of concentric rings. The loss of symmetry at thecenter of the reconstructed plane can be attributed tothe role played by the crystal geometry when theoverall orientation of the convection pattern is de-cided. The effect of crystal shape is negligible whenthe size is small (for example, at t � 2 h) but issignificant when the crystal grows in size with a well-defined morphology (Fig. 10). In addition, denselyspaced contours in the core and the bulk reflect thepresence of high-concentration gradients in thegrowth chamber.
4. For planes located away from the crystal �y�H� 0.60�, the effect of crystal geometry on the con-vective field is diminished. The presence of a darkcentral core (corresponding to a higher level of super-saturation) surrounded by a brighter region showsthat high-concentration gradients prevail on the crys-tal sides in comparison with its upper face.
The stable convection regime of crystal growth yieldsthe highest quality of the crystal in terms of symme-try of the faces and its transparency, along with the
fastest growth rate. The morphology of the KDP crys-tal is pyramidal, as shown in Fig. 10 in a photographof KDP crystals grown in the diffusion-dominated aswell as the convection regimes.
6. Conclusions
Concentration distribution around a KDP crystalgrowing from its aqueous solution is reported. Mea-surements have been carried out using a mono-chrome schlieren technique. Images recorded atvarious view angles have been analyzed using theconvolution backprojection algorithm. A suitable ex-trapolation scheme has been employed to generateinformation about the entire measurement volumeusing the partial projection data recorded throughexperiments. Steady crystal growth experimentshave been conducted in a regime that is dominated byconvection. The following conclusions emerge fromthe present study:
1. Tomographic reconstruction applied toschlieren data revealed solute concentration distri-bution over horizontal planes above the growingcrystal.
2. The concentration field in the vicinity of a crys-tal growing in a diffusion-dominated regime showedloss of symmetry in the initial phase. Symmetry wasrestored at longer times indicating a uniform deposi-tion of solute on the crystal faces. The field in the bulkof the solution was entirely axisymmetric in the ini-tial stages of the growth process.
3. With an increase in the crystal size, the convec-tion current was seen to increase in strength andthus determine the overall transport of solute fromthe bulk of the solution to the crystal faces. The re-sulting growth rate was higher in comparison withthe diffusion regime. Nearly symmetric distributionof concentration was realized in the stable growthregime in regions away from the crystal.
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