OPTICAL DIAGNOSTIC AND MODELING SOLUTION GROWTH PROCESS OF SODIUM CHLORATE CRYSTALS

Applied Mathematics and Mechanics (English Edition), 2006, 27()177 -1184⑥Editorial Committee of Appl. Math. Mech, ISSN 0253 4827OPTICAL DIAGNOSTIC AND MODELING SOLUTIONGROWTH PROCESS OF SODIUM CHLORATE CRYSTALS *WANG Tao (王涛)，DUAN Li (段俐)(National Microgravity Laboratory, Institute of Mechanics, Chinese Academy of Sciences,Beijing 10080, P. R. China)(Communicated by HU Wen. rui)Abstract: Both a real time optical interferometric experiment and a numerical sim-ulation of two- dimension non-steady state model were employed to study the growthprocess of aqueous sodium chlorate crystals. The parameters such as solution concen-tration distribution, crystal dimensions, growth rate and velocity field were obtained byboth experiment and numerical simulation. The infuence of earth gravity during crystalgrowth process was analyzed. A reasonable theory model corresponding to the presentexperiment is advanced. The thickness of concentration boundary layer was investigatedespecially. The results from the experiment and numerical simulation match well.Key words: crystal growth; interferometry; numerical simulation; concentration bound-ary layerChinese Library Classification: 07812000 Mathematics Subject Classification: 74N05; 74N25Digital Object Identifter(DOI): 10.1007/s 10483 006-0904-1IntroductionCrystal growth is a procedure of phase changing. It is infuenced by kinds of physicalphenomena. Since structure details of crystal growth process from either melt or solution arestill not fully understood, a complete convincing theory model is not advanced by now. Thequality and shape of crystals in growth process are known to be affected by many factors, suchas impurities, steps bunching, and the transport process of difusion and convection. Furthermore there is a lttle diference in microgravity condition between that in terrestrial gravity. So,more and more researches in this field are carried out to discover the physical essentials and toimprove the growth model.According to the larger crystal scale, ground-based inorganic crystal growth experimentis much more facile for optical observing than that of protein crystal. And the experimentshave obtained many significant results. Onuma and Tsukamoto developed the phase shift opticinterface method with high resolution[1,2), and Duan et al. improved it to show the concentrationtransferl3- 5]. This Mach-Zehnder interferometry system can be used for real time observation ofconcentration field and velocity field. Then experiment data are processed by an image processsystem to obtain the crystal growth rate and the thickness of concentration boundary layer.In this paper, a same Mach- Zehnder interferometry system is employed to observe the growthprocess of NaClO3 crystals.Modeling the process of crystal growth is an attra中国煤化工，only dynam-icists but also geologists, biologist (especially in biostrMHC N M H Gts[6-i0]. Many* Received Nov.1, 2005; Revised Jun.1, 2006Project supported by the National Natural Science Foundation of China (No.10432060)Corresponding author DUAN Li, Associate Professor, Doctor, E mail: duanli@imech.ac.cn1178WANG Tao and DUAN Ligrowth procedures in diferent environments including melt growth, solution growth and Va-por diffusion are widely discussed recentlyl7 -9,11. In present paper, a two-dimension linearconvection-diffusion model is advanced如match the aqueous NaClO3 crystallization exper-iment. The crystallization experiment and the numerical simulation are both discussed inpresent paper to obtain the mechanical essentials of the growth process under earth gravity.According to the experiment results, the reasonability of the growth model is validated.1 Theoretical Analysis and ModelThe terrestrial crystal growth process is inevitably affected by earth gravity. The gradient ofsolution concentration brings on layer deposition, which turns into buoyancy convection underearth gravity. As the crystal growth process is affected by the buoyancy convection, it may causecrystal defects in dimension, integrality, optical uniformity, etc. The process of crystal growth isconsidered as a combination of the bulk transport and the interface kineticsl6,10,12. In the bulktransport procedure solute is transported to the growth interface from the bulk, while in theinterface kinetics procedure the solute on the growth interface moves to the kinks to enter thecrystal lattice. Bulk transport is controlled by the difference of bulk concentration C and surfaceconcentration Cs, and interface kinetics is driven by the difference of surface concentration Cand equilibrium concentration Ceq. The two procedures couple to conduct the whole growthprocess. In extreme situation of absolute bulk transport surface concentration approach toequilibrium concentration: Cs→Ceq, while in the opposite absolute interface kinetics situationCg→Cl6,8,10]. A simplified two-dimensional model is employed to simulate the non steadyprocess of the terrestrial crystal growth in Boussinesq approximation. A coordinate systemwhich fixed on the upper crystal surface is considered. The process is governed by the followingequations[67,10I:8u、 8v= 0,(1)8x 838u)u8P-+ v(2)Ht'的y8路+影),0v 8u .. 8v__ 8P82v开+u元+"可=- 8+(02+ 82) -gBc(C-Co),(3)oC、 8C. 8C182C 82C\+u+Ua=D0x2+oy)(4)Boundary conditions at the crystal growth cell wall:8C(5)anu|r=-a,a=0，v|x=-a,a=0.(6)Boundary conditions at the solution-atmosphere interface (free interface):中国煤化工P= Po,(7)C = Ceva,MYHCNMHGv|y=l= 0.(9)Optical Diagnostic and Modeling Crystal Growth Process1179Boundary condition at the growth interface (Fick's law):D咖= K(C -C).(10)Initial condition;C=Co..(11)Here u, U are the solution velocity of x，y-direction, respectively, v is the kinematic viscosity,βc is the cofficient of solutal expansion, D is the solute difusivity, n is the normal directionof the cell wall, P is the atmospheric pressure, Ceva is the steady evaporation concentration ofthe free surface, and Co is the initial solution concentration.Commercially available code FLUENT is used for numerical simulation. A 96 x 96 meshis used to correspond an area of 12 mmx 12 mm of actual dimension. The 4 rows mesh nearthe crystal are fined (Fig.(1)). The fner mesh shows a much clearer result of velocity field andconcentration distribution near the interface. At the side-wall and the bottom of the growthcell, it is given a non- diffusive fux boundary condition (Eq.(5)). The upper surface, which isa free evaporation surface, is given a fixed evaporation surface concentration (Eq.(8)) as theenvironment parameters keep invariable such as temperature and humidity. The seed crystal of3 mmx 1 mm dimension is placed on the bottom center of the growth cell. Boundary conditionat the growing interface is given by the Fick's law (Eq.(10)). This model is a simulation of theactual experiment with similar geometry scales and the same boundary and initial conditions.So the results obtained from the numerical simulation are comparable to those of the actualexperiment. Thus the physical process of growth can be properly investigated.Fig.1 Mesh used in simulation2 Experiment of Crystal Growth and ResultsA set of Mach Zehnder interferometer system (Fig.(2)) is employed in the experiment toobtain the concentration gradient of the solution fuid field in the growth process. In the Mach-Zehnder interferometer system, a beam of laser passes through a group of lens (a) to forman expended parallel light beam, and then the beamlit hv P cnlittor h) into two beams.One beam of laser, which crosses the crystal growth (中国煤化ject beam, andthe other is used as the reference beam. The two beaHCNMHGtandformaninterferometric fringe pattern, which carries out the concentration gradient. Then fringe imagesare received by a CCD camera (f) and recorded by computer with a Matrox image capture card.And later these images are processed by computer.1180WANG Tao and DUAN Li@←t母a: Lensb, d: Splitterc: Growth celle: PZT phase shifterf: CCD cameraFComputerg: Drive systemh: Image systemFig.2 Schematics of experiment setThe aqueous growth of sodium chlorate crystals is studied in present experiment. The48.0% concentration of NaClO3 solution prepared at 18 °C is injected into the crystal growthcell of 12 mmx9 mmx10 mm dimension, which is made of optical glass. A seed crystal of2.78 mmx 1.32 mmx0.72 mm dimension is placed on the bottom of growth cell. The uppersurface of the crystal is face (0 0 1). The left and right faces are faces (0 1 0). Environmenttemperature is kept at 18°C invariably during the whole process of experiment. At the verybeginning, the solution is under saturated, but due to the dissolution and evaporation, thesolution concentration increases. In few minutes the solution is saturated, and the dissolutionstopped. As the evaporation continues, the solution gradually becomes supersaturated. Thussolute near the crystal is adsorbed to the growth interface, and enters the crystal lattice aftera complex interface kinetic process. As the concentration in adjacent volume decreases, thedensity of this part of solution is lower than the bulk. So under the earth gravity, a plumbflow from crystal surface to the top of solution is formed. This is the buoyancy convection as aresult of concentration change under gravity. In present experiment, the interferometric fringesrelated to the concentration gradient of the solution are shown clearly in the captured images.After 6 hours' growth, a NaClO3 crystal of 4.56 mmx2.30 mmx 1.18 mm dimension is obtained(Fig.(3)).t= 125 mint= 250 min1= 340 minFig.3 Evolution of interferometric fringe中国煤化工Interferometric fringes from experiment image datajY片C N M H Qn distributionof the fAuid field. As shown in Fig.(4), a four-step phase snit Tmage rig.4(a) is interpretedin phase calculation and phase unwrapping to obtain the solution concentration distribution(Fig.4(b)) at a certain time. In Fig.(4), the experiment time is t = 240 min.Optical Diagnostic and Modeling Crystal Growth Process1181The concentration distribution obtained from phase calculation and phase unwrapping alsoshows the difusion boundary layer. These images demonstrate that the concentration near thegrowth interface decreases rapidly to form a growth boundary layer. As shown in Fig.4(c), ac-cording to the concentration curve at Z = 3.15 mm, the thickness of the concentration boundarylayer is approximately 200 pm.(x) Four -step phase shift interferometric fringes(b) Concentration distributionin crystal growth process5251ξ4847404:0.00 0.54 1.08 1.61 2.15 2.68 3.22 3.75 4.29 4.82 5.36X/mm .(c) Concentration distribution curve at borizontal line Z = 3.15 mmFig.4 Concentration distribution at t= 240 min3. Numerical SimulationNumerical simulation of the non-steady model introduced in section 1 is used to calculatethe fuid field. of crystal growth process. The solution parameters at t = 240 min are taken asa sample. The velocity field is shown in Figs.5(a) and 5(b), and the concentration distributionin Figs.5(c) and 5(d). As a result of the earth gravity, plumb flow of buoyancy convection isformed in the fuid field Fig.5(a). At the area closed to face (0 0 1), Alow moves vertically tothe upper evaporation surface. The flow near the crystal is faster than the other part of theAlow field. The crystal is enwrapped by the fow near th中国煤化工Concentration near the growth interface reducesentration Cg.Thus a concentration boundary layer is formed. MeantYHCNMHGfthegrowthcell, the concentration changes rapidly from the invariable evaporation concentration to the1182WANG Tao and DUAN Li(a) Velocity field(b) Velocity field near growth interface(c) Concentration distribution(d) Concentration distribution near growth interfaceFig.5 Velocity field and concentration distribution at t = 240 minsolution concentration as well. In the other part of the solution, the concentration remains ata certain level, although the buoyancy convection changes the concentration distribution.Velocity field and the concentration distribution are calculated by the same simulation modelatt= 15 min and t = 50 min (Fig.6). And the results of calculation is synthetically analyzedto study the evolvement of velocity field and concentration field in the entire process of crystalgrowth together with those at t = 240 min. It is demonstrated that the solute in the adjacentvolume of the growth interface is frstly adsorbed at the very beginning, and a plumb convectionis formed as the density reduced gradually. After a certain period of growth, the velocitydistribution and the concentration distribution are prone to be stabile until the end of thegrowth process.Considering the calculation of concentration distribution (Figs.5(c),5(d),6(c),6(d)), the bo-undary layer near face (0 1 0) changes slightly due to the infuence of the convection. And thethickness remains at a certain value. At the time t中国煤化工F the boundarylayer is 166 pm, which approximately matches that of(Fig.(7)). Allthe few main steps of the growth process occur in theMHC N M H Gyer, includingthe transportation of solute from solution to the interface and the entrance into crystal lattices.Thus the research of concentration boundary layer is of practical significance. In following ex-periments, equipments with even higher resolution are required to obtain more effective results.Optical Diagnostic and Modeling Crystal Growth Process1183(@) Velocity field at t = 15 min(b) Velocity field at t = 50 min(c) Concentration distribution at t = 15 min(d) Concentration distribution at t= 50 minFig.6 Evolution of velocity feld and concentration distribution50.049.5十 Simulation一Experiment。48.5-i 48.0-47.547.0-0.05 0 0.05 0.10 0.15 0.20 0.25 0.30x1 mmFig.7 Concentration distribution curve of simulation results and experiment data at thehorizontal line Z = 3.15 mm, t= 240 min (X = 0 is the growth interface)4 Conclusions中国煤化工MHCNMHGThe experiment data provide evolvement of several pnysIcal parameters aurng the growthprocess such as the crystal dimension, concentration distribution, thickness of the concentrationboundary layer, etc. And the numerical simulation of the two-dimension non- steady model1184WANG Tao and DUAN Ligives the same parameters by calculation. Comparing the two methods, the simulation resultsmostly match those from experiment. Thus this model does correspond to the actual situationof present experiment, especially concerning the parameters of the solution fuid field. As inthe inorganic crystal growth process, the infuence of nonlinear process is not quite obviousbecause of large crystal scale and high growth rate, the nonlinear interface kinetic process isnot discussed in this model. In further discussions of protein crystal growth process, when theinfuence could not be neglected, the nonlinear kinetics may play a more important role to thenumerical simulation. To advance a proper assumption to the interface nonlinear kinetics ofprotein crystal growth process is the main objective of further researches.References{1] Onuma K, Tsukamoto K, Sunagawa I. Role of buoyancy driven convection in aqueous solutiongrowth: a case study of (BaNO3)2 crystal[J]. 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