Indian Journal of Pure & Applied Physics Vol. 43, January 2005, pp. 24-33

Simulation and experimental verification of solutal convection in the initial stages of crystal growth from an aqueous solution

Sunil Verma*, Atul Srivastava+, Vivek Prabhakar+, K Muralidhar+& V K Wadhawan*

*Laser Materials Division, Centre for Advanced Technology, Indore 452 013 +Department of Mechanical Engineering, Indian Institute of Technology, Kanpur 208 016

Email: sverma@cat.ernet.in, Fax: 0731-2488650

Received 12 July 2004; revised 6 August 2004; accepted 25 October 2004

Three-dimensional, unsteady numerical simulation of transport phenomena during the initial stages of growth of the KDP crystal from its aqueous solution and comparison with experiments have been presented. The convective field in the solution is set-up due to density differences between the supersaturated solution in the bulk and the saturated solution in the vicinity of the crystal. The flow field in the solution has been visualized using a shadowgraph technique. Numerical as well as experimental results of the growth under free convection show that convection is initially set-up in the vicinity of the crystal, while the solution in the rest of the growth chamber is relatively stagnant. The initial flow patterns show symmetry to a great extent. The influence of the crystal rotation on the concentration field has been additionally studied through numerical simulations. When the crystal is rotated, the gradients in concentration between the region near to the crystal and far away from it are high. This is a desirable requirement for a higher crystal growth rate. With the passage of time, the length scales of convection span the growth chamber, and a purely unsteady flow field is established. The numerical simulation could not be continued for such long durations owing to excessive requirement of computational resources. However, a good agreement between simulation and experiments for short times indicates that the imaging of the flow field provides a useful strategy for continuous monitoring of crystal growth.

[Keywords: Computer simulation, Convection, Heat transfer, Crystal growth from solution, Potassium dihydrogen phosphate, Shadowgraphy]

IPC Code: C3 0B 7/00

1 Introduction Advancement in the field of photonics critically

depends upon the progress made in science and technology of crystal growth. In the last decade giant leaps have been taken in this direction, primarily due to the contribution of inter-disciplinary research teams, particularly dealing with fluid dynamics and optics. Fluid dynamics simulation of the transport phenomena and its visualization by optical techniques can provide critical insights into the complex nature of flows occurring during growth. It can improve the related instrumentation and provide a push to the technology of growing crystals.

Transport phenomena, including heat and mass transfer, and fluid flow strongly influence the growth process1-5. It affects the growth rate and the quality of the growing crystal in terms of the compositional homogeneity, morphological stability and the defect structure. There are several factors that affect the

convective field simultaneously, such as gravity, surface tension, and forced convection. Hence, the flow structure is highly complex. Carefully planned simulations and experiments are needed to study them. Rosenberger and Muller6 have discussed rigorous modeling of the transport phenomena requiring simultaneous solution of the conservation equations of momentum, mass, species and energy. They opined that an idealized modeling approach yields very little insight into the process and can only be followed for simplified geometry, whereas the crystal shapes encountered in practice are highly complex. They concluded that several approximations are necessary to bring the idealized model close to realistic conditions. These remarks were made when the computational resources were primitive; however, it would not be inappropriate to add that they are valid at present, since only a few research groups, having access to supercomputing resources have successfully

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performed 3D time dependent simulations of mass transfer and fluid flow for a few important crystals, such as KDP and KTP, for the actual growth geometries7-11. Such isolated advancements have pushed the state-of-the-art of the crystal growth science to new heights.

Optical techniques as measurement tools have seen resurgence in engineering research because of the availability of lasers as sources and computers for data acquisition and processing. They have the specific advantage of being non-intrusive in nature, thus leaving the field undisturbed. In addition, they have the potential of providing inertia-free and whole-field measurement of the process. In the flow visualization research, classical optical techniques of interferometry, schlieren and shadowgraphy have been extensively used12. With the availability of several newer techniques such as the laser Doppler velocimetry, the particle image velocimetry, the speckle and moiré interferometry and scattering based techniques13, the spectrum of applications has considerably widened.

In the present paper, we have used the shadowgraph technique to visualize the convective field during crystal growth from its aqueous solution. In addition, the CFD simulation of solute transport occurring around a KDP crystal in the initial stages of growth has been performed. A general-purpose commercial code FLUENT-5.5 has been used for this purpose. Simulations for very long durations could not be accomplished due to severe requirement of computational time, even on powerful workstations.

2 Simulation Details In order to have a meaningful comparison between

the simulation and the experimental results, the simulation geometry is chosen to be close to the experiment. The growth chamber is a horizontal cylinder of diameter 55 mm and length 145 mm. The growing crystal is a vertical cylinder of diameter 7.5 mm and length 15 mm placed in the middle of the cell. The schematic diagram of the simulation geometry is shown in Fig. 1.

The FLUENT module available to the authors did not have provisions for mass transfer calculations. For this reason, an equivalent heat transfer model has been solved to study the evolution of buoyancy-driven convection flow in the fluid domain. The driving force for convection in the heat transfer model is the temperature difference between the crystal and the solution in the far field. It is equivalent to the density

difference between the far field supersaturated solution and the saturated solution adjacent to the crystal. In non-dimensional form, it is represented by the solutal Grashof number (thermal Grashof number in case of heat transfer model), namely the ratio of the buoyancy force in a non-isothermal liquid and the viscous dissipative force, and measures the strength of the buoyancy-driven convection in the crystal growth process. The equivalence of the heat and mass transfer models can be expressed by equating the thermal and solutal Grashof numbers:

33

2 2

( )( ) β −β Δ=

ν νC supersat satT

water KDP

g C C Lg T L … (1)

where g is the acceleration due to gravity, βT and βc are the thermal and solutal coefficients of volume expansion, and νwater and νKDP are the kinematic viscosity of the water and KDP solution respectively. From Eq. (1), an equivalent temperature difference corresponding to the difference of concentration between the saturated and supersaturated solutions of KDP at an average temperature of 50°C has been calculated using the solubility and super-solubility data reported in literature14. On the basis of the temperature difference obtained, the following boundary conditions are enforced: the walls of the growth cell are adiabatic; the crystal boundaries are at a specified temperature of 340 K; and the temperature of the solution is 280 K. The property values have been evaluated at atmospheric pressure.

Numerical simulation for studying the effect of free convection on the thermal (and hence concentration) fields inside the growth chamber was first performed using FLUENT-5.5. It is a commercial package for modeling fluid flow and heat transfer processes. In the present work, the flow field considered was laminar and the computations were performed to determine the unsteady convection patterns of the three-dimensional convective field. The flow was taken to

Fig. 1⎯Schematic drawing of the simulation geometry

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be incompressible, with buoyancy accounted for via the Boussinesq approximation. The governing equa-tions are mathematically represented by the Navier-Stokes equations, the continuity equation, and the energy equation. An implicit segregated solver on an unstructured grid is used to solve the governing equations. The pressure equations are body-force weighted while the momentum and the energy equations are discretised using the first order upwind scheme. The SIMPLE algorithm is used to resolve the pressure-velocity coupling15. All the velocity compo-nents are initialised to zero before the start of the iterations. Convergence criteria of 10-3 for velocity and 10-6 for energy equation are employed to stop the iterative procedure.

The effect of forced convection on the evolution of thermal (and concentration) fields inside the growth chamber has also been simulated. Rotation rates of 10 and 50 rpm were provided to the crystal, and the evolution of the scalar fields was determined at different time instants. The temperature maps were obtained at three different depths inside the growth chamber, namely at L/6, L/4 and L/2, where L being the length of the cylindrical growth cell.

3 Apparatus and Instrumentation 3.1 Crystal growth apparatus

A special growth cell was designed and fabricated for performing the experiments. It was a double walled chamber made of glass and has a horizontal cylindrical geometry. The cavity enclosed between the walls of the two concentric cylinders was filled with thermostated water, maintained at the desired temperature by the temperature controlling circuitry. It helps in maintaining the temperature of the KDP solution within ± 0.01°C. In order to facilitate unhindered passage of the laser beam through the growth cell, it has been provided with quartz optical

windows on either side of the cylindrical cell. Specific locations are provided on the growth cell for introducing a temperature sensor and a seed holder. The stability of the solution temperature is crucial to conduct experiments with repeatability, and to avoid the problem of spurious nucleation. In the present work, a PID based temperature controlling instrumen-tation [Eurotherm] was used. The signal from the temperature controller is dependent on the output of a comparator circuitry, which continuously compares the set-point temperature with the process temperature.

3.2 Optical set-up and data acquisition instrumentation

Figure 2 shows the schematic layout of the optical and the image acquisition instrumentation. A He-Ne laser of 5 mW power is expanded and collimated to a beam diameter of 10 mm by a beam expander. The collimated beam passes through the crystal growth chamber in which the KDP crystal is growing. The beam emerging out of the exit window of the growth cell falls onto a screen resulting in the shadowgraph image. In turn, it is continuously recorded using a video camera (Sony Handycam 360X) interfaced to a computer through a 1024×1024, 25 Hz frame grabber card.

3.3 Crystal growth

In order to grow a KDP crystal from its aqueous solution, a saturated solution was prepared using chemicals of 99.9% purity and de-ionized water having an electrical resistivity of 18 MΩ-cm. To reduce the problem of spurious nucleation, the solution was filtered using a special membrane filter, 0.02 μm pore size, and then over-heated for 24 hours to dissolve any possible clusters. The growth was initiated by using a small transparent crystal of KDP of size 3×3×3 mm3 in the top hanging geometry. The

Fig. 2⎯Optical and image acquisition instrumentation

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temperature of the solution was lowered after appropriate time intervals in small steps of 0.2°C/hr. The growth was performed over a temperature range of 48 to 36°C. The quality of the crystal grown in the laboratory was uniformly found to be high.

3.4 Shadowgraph experiments

The convective field around the growing KDP crystal was continuously monitored using the shadow-graph technique. In the presence of the growing crystal, the KDP solution is a field of varying refractive index, and the light rays passing through it are refracted out of their original path. This causes a spatial modulation of the intensity distribution with respect to the original intensity on the screen. This results in the shadowgraph image were continuously recorded using the instrumentation described in Section 3.2.

4 Shadowgraph Analysis

Consider the aqueous solution as a medium with a refractive index n that is a function of the three space co-ordinates: ( ), ,n n x y z= . We are interested in obtaining the governing equation of the shadowgraph process by tracing the path of light rays as they pass through this medium. According to Fermat's principle the optical path length between the incident and the exit points of the ray has to be a minimum. This condition in a parametric form involving two functions ( )x z and ( )y z can be written as16:

2 2( , , ) ' ' 1 0e

i

z

z

n x y z x y dz⎛ ⎞

δ + + =⎜ ⎟⎜ ⎟⎝ ⎠∫ … (2)

where the primes denote differentiation with respect to z. Application of the variational principle to Eq. (2) yields two coupled Euler-Lagrange equations, which can be written in the form of following differential equations for ( )x z and ( )y z :

( ) ⎟⎠⎞

⎜⎝⎛

∂∂

−∂∂

++=znx

xnyx

nzx '''11)('' 22 … (3)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

++=zny

ynyx

nzy '''11)('' 22 … (4)

The solution of these coupled differential equations with the knowledge of the refractive index provides the measure of the deviation of the ray as it passes

through the refractive medium. The deviation of the rays from their original paths in passing through the in-homogeneous medium results in the change of intensity distribution on the screen as compared to the original distribution when the disturbance is not present in the beam path. The shadowgraph technique measures this change in the intensity distribution. The intensity at point ( ),s sx y on the screen is a result of several beams moving from locations ( ),i ix y and getting mapped onto the point ( ),s sx y on the screen. The intensity at point ( ),s sx y can be written as:

( )( , )

( , )( , ),

( , )i i

o i is s s

x y s s

i i

I x yI x yx yx y

=∂∂

∑ … (5)

where 0I is the intensity at a particular point ( ),i ix y on the screen in the absence of the field of disturbance, sI is the intensity at point ( ),s sx y due to intensities ( )0 ,i iI x y of all the points ( ),i ix y which are mapped onto ( ),s sx y in the presence of the field of disturbance. The denominator in Eq. (5) is the Jacobian of the mapping function of ( ),i ix y into

( ),s sx y . It geometrically represents the ratio of the area enclosed by four adjacent rays before and after passing through the in-homogeneous medium. In the absence of the in-homogeneous field, such an area is a rectangle, that transforms to a deformed quadrilateral when imaged onto a screen in the presence of the in-homogeneous field. The summation in Eq. (5) extends over all the ray points ( ),i ix y that get mapped onto ( ),s sx y on the screen and contribute to the intensity at that point. Under the assumption of infinitesimal displacements, the Jacobian can be assumed to be linearly dependent on them. Thus the higher powers of ( )s ix x− and ( )s iy y− , and their product can be neglected. Under this linearizing assumption the governing equation for the shadowgraph process can be shown to be16:

( ) { }2 2

2 2 ln ( , )o s

s

I I L D n x yI x y

⎛ ⎞− ∂ ∂= × +⎜ ⎟∂ ∂⎝ ⎠

… (6)

where L is the spread of the field of disturbance, D is the distance between the exit surface of the growth

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cell and the screen, and ( ),n x y is refractive index at the location ( ),x y .

The Poisson Eq. (6) has been solved numerically for two representative shadowgraph images grabbed at the start of the experiment and after a few hours of growth. The solution of Eq. (6) yields the instan-taneous 2-D refractive index distributions integrated along the length of the growth cell. A computer code has been developed for solving the Poisson equation and validated against analytical examples. The output generated by the solver is a matrix giving the refractive index at each node point of the starting grid.

The relationship between the refractive index and the concentration of the KDP solution saturated at different temperatures is well documented in literature14.

5 Results and Discussion

The simulated concentration field around a KDP crystal growing under free convection at time instants of 25, 50, 100, 150, 200, 300, 400 and 600 seconds is shown in Fig. 3 (a-h) respectively. We see that as the time progresses, the free convection effects spread through the growth chamber. The shades of gray represent the absolute value of concentration, with

Fig. 3(a-h)⎯Simulated concentration fields after 25, 50, 100, 150, 200, 300, 400 and 600 seconds of growth under free convection

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darker regions being at higher concentration as compared to lighter regions. It is observed that the effect of the concentration gradient, has not spread uniformly inside the growth chamber, and there are small differences in the values of concentration near to the crystal and far away from it.

An experimental shadowgraph image of the concentration field during the initial stages of growth under free convection is shown in Figure 4. Here, the difference in the concentration field is visible in the intensity distribution near the edges of the crystal and far away from it. The region of depleted concentration is visible as a bright halo in the immediate vicinity of

the crystal. An additional verification of the same result is shown in Fig. 5, which is the refractive index contour map obtained by solving the governing equation of the shadowgraph process for the image in Fig. 4. Here two well-separated regions of refractive index are visible, one enveloping the crystal and other away from it, showing that the gradients in the concentration are localized in the initial stages.

A shadowgraph image of buoyancy driven convec-tion after 4 hours of growth is shown in Fig. 6. We observe that the convective activity is very vigorous, as is visible in the high contrast plumes near the crystal edges. The corresponding refractive index contour map obtained by solving the governing equa-tion of the shadowgraph process is shown in Fig. 7. Here, we see that the concentration field has fully spread inside the growth chamber and the difference in the value of refractive index near the crystal and far away from it is very high. It indicates the presence of high concentration gradients throughout the solution. Simulations of the concentration field for such large durations of the growth were not possible using FLUENT because of the severe requirements of computational resources and time.

The effect of forced convection is studied by providing rotation to the crystal at two different rates of 10 and 50 rpm. Fig. 8(a-h) are the simulated concentration field images after 25, 50, 100, 150, 200, 300, 400 and 600 seconds of growth when the crystal is rotated at 10 rpm. The comparison between free and forced convection images corresponding to the

Fig. 5⎯Refractive index field corresponding to shadowgraph image in Fig. 4

Fig. 4⎯Shadowgraph image in the beginning of growth under free convection

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same instants of time reveals that the forced convection has resulted in uniform mixing in a much shorter time. This can be seen in the form of reduced concentration difference between the region close to the crystal and far away from it, implying smoother concentration gradients. In order to visualize the effect of the intensity of forced convection, simula-tions were performed for 50 rpm. The simulated

concentration fields are shown in Fig. 9(a-h) for the respective time instants. Here it is seen that as the intensity of forced convection increases the uniformity of the concentration field is achieved even faster. This is a desirable feature during crystal growth, as it results in a uniform growth rate of all the crystal faces. In addition, rapid replenishment of the solute depleted region by the solute rich solution is accomplished. This is particularly necessary for the fast growth technology of large optical crystals. Although solute uniformity (smaller gradients spread over a larger region) improves as we increase the strength of the forced convection, it has to be carefully optimised to avoid turbulent flows that are counter-productive and can result in stray nucleation away from the crystal surface.

6 Conclusions

We have studied the evolution of concentration field during the initial stages of growth of a KDP crystal, under free as well as forced convection conditions. The salient conclusions of our study are:

(a) In the initial stages of growth under free

convection the concentration gradients are localized in the vicinity of the growing crystal. The quantitative analysis of the experimentally recorded shadowgraph image also reveals the above observation.

Fig. 6⎯Shadowgraph image after 4 hours of growth showing vigorous convective activity. The plumes seen descending from the crystal edge actually move upwards at a distance slightly away from the edge. This could not be captured owing to small dimension of the expanded beam.

Fig. 7⎯Refractive index field corresponding to the image in Fig. 6

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(b) After a few hours of growth, the convective plumes were observed to be vigorous indicating the presence of high concentration gradients in the vicinity of the crystal as well as in the bulk of the solution. This was verified through quantitative analysis of the shadowgraph image grabbed after a few hours of growth.

(c) Results of the numerical simulations performed to study the effect of forced convection on the concentration field show that the mixing of the solution is achieved very fast, and the uniformity of the solution around the crystal is improved as compared to the free convection case.

Fig. 8(a-h)⎯Simulated images of the concentration fields after 25, 50, 100, 150, 200, 300, 400 and 600 s respectively when the crystal is rotating at 10 rpm

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(d) As the intensity of the forced convection is increased, the uniformity of the solution concentration is further enhanced. Consequently, a diminished difference in the solution concentration near and far away from the crystal is observed. This is a useful feature for crystal growth as it leads to uniform growth rate of all the faces. Rapid replenishment of the solute in the salt depleted regions leading to an enhanced growth rate is a second advantage of growth under forced convection.

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Fig. 9(a-h)⎯Simulated images of the concentration field after 25, 50, 100, 150, 200, 300, 400 and 600 seconds, respectively, when the crystal is rotating at 50 rpm

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