Download Comparison of Interferometry, Schlieren and Shadowgraph Techniques in Crystal Growth and more Slides Mechanical Engineering in PDF only on Docsity!
Module 5: Schlieren and Shadowgraph
Lecture 31: Results and discussion related to crystal growth (part 1)
The Lecture Contains:
Convection Phenomena During Crystal Growth
Comparison of Interferometry, Schlieren and Shadowgraph in a crystal Growth Experiment Convection Patterns Comparision of the Three Techniques Influence of Ramp Rate and Crystal Rotation on Convection Patterns Ramp Rate of Ramp Rate of
Module 5: Schlieren and Shadowgraph
Lecture 31: Results and discussion related to crystal growth (part 1)
CONVECTION PHENOMENA DURING CRYSTAL GROWTH
For a purely buoyancy-driven growth process, the driving potential of flow is the maximum concentration difference occurring in the solution. In addition, the strength of convection is also governed by the length scale, typically the size of the growing crystal. With time, the solution is depleted of the salt, though there is an increase in the length scale related to the change in size of the crystal. Jointly, the strength of convection can increase with time, till the solution in the growth chamber is fully depleted of salt. This state is characterized by stable stratification of the solution, the convection currents diminishing in strength to negligible levels. The growth of the crystal practically stops at this stage. Growth can be resumed when the grown crystal is immersed in a fresh supersaturated solution and the ramp rates are re-introduced. The resulting convection patterns would be different from the first stage because of a change in the crystal size. When the crystal is imparted rotation, velocities are created in the angular direction in the horizontal plane, in addition to the buoyancy-driven motion in the vertical plane. However, the two motions are interlinked through the radial component of velocity. The linkage is such that rotational motion leads to Coriolis forces that re- direct fluid motion. However, homogenization of the solution is the dominant factor that reduces concentration gradients, diminishes the driving potential and hence suppresses fluid motion arising from buoyancy. The critical speed at which buoyancy is practically suppressed will depend on the crystal size and overall concentration difference available in the solution. The resulting solutal concentration distribution at the surface of the crystal influences the growth rate and quality.
The process of solute deposition leading to crystal growth occurs on a hierarchy of length and time scales. At the small scale, solute particles arrange themselves as a part of the crystal structure. The pyramidal structure seen at later stages of growth is initiated at this point. The experiments conducted in the present work do not yield information on this aspect of the growth process. At the larger scale (the length scale of the crystal itself), concentration gradients are set up that feed solute to the crystal. These gradients naturally control the rate of crystal growth. The uniformity in distribution of the concentration gradients determines the crystal quality. The present research aims at investigating physical mechanisms at the scale of the crystal in terms of the solutal concentration distribution.
Figure 5.16: Time sequence of the evolution of interferograms around the
growing crystal. (a-d) Infinite Fringe setting; (e-h) Wedge fringe setting. The
initial crystal size in the infinite fringe setting is greater than in the wedge
fringe setting. Large fringe slopes in (e-h) very close to the glass rod are
possibly distortions. The opposing fringe curvatures above and below the
crystal in (h) show a lighter and a denser solution formed by stratification
Module 5: Schlieren and Shadowgraph
Lecture 31: Results and discussion related to crystal growth (part 1)
Figure 5.16 shows the formation and time-evolution of fringes in the infinite and wedge fringe settings of the interferometer around the growing crystal. The first image of the transient shows the appearance of fringes due to crystal dissolution. This process increases the local density, causing the solution to descend vertically downwards. Hence, the fringe displacement is also in the downward direction (Figure 5.16(a)). Large concentration gradients in the vicinity of the dissolving crystal give rise to time-dependent fluid motion as well. A second factor contributing to unsteadiness is the change in the crystal geometry from rectangular to prismatic. As the crystal attains thermal equilibrium on one hand and its natural shape on the other, the concentration gradients gradually diminish. During a time period of 15-35 hours, the gradients were small enough to produce only a single visible fringe adjacent to the growing crystal (Figure 5.16(b)). The fringe was found to be stable with respect to time indicating uniform deposition of the solute around the growing crystal. Experiments in the wedge fringe setting of the interferometer reflect identical trends. The horizontal fringes deform vertically downwards (Figure 5.16(e-f)), and is followed by a phase when they are practically straight. Near the crystal, the solution is practically saturated, while it is supersaturated in the far field. The wedge fringes get displaced in regions of a large change in concentration with respect to the bulk of the supersaturated solution. With further cooling, the solution in the near-field becomes supersaturated with reference to the new temperature, additional salt deposits on the crystal, and the growth process is once again initiated. This leads to concentration gradients adjacent to the crystal, and a continuation of buoyancy-driven flow. In the time frame of 20-30 hours after the insertion of the KDP seed, the infinite and the wedge fringes showed considerable symmetry as well as stability in time, ensuring uniform growth on all the faces of the crystal. This time duration may be called the stable growth regime of the crystal.
With an increase in the crystal size, the influence of even mild concentration gradients is strengthened, increasing the fringe deformation. Over a longer duration of the experimental run time (> 50 hours), the solution is found to be layered (stratified) with respect to density (Figures 5.16(c-d) in the infinite fringe setting and Figures 5.16(g-h) in the wedge fringe setting). This is understandable because crystal growth takes place from a fixed volume of the solution in the growth chamber, and with time, the solution is increasingly depleted of salt. The density inversion suppresses convection to a point where the increase in the crystal size is negligible. The downward movement of these layers of constant concentration is driven by molecular diffusion, and contributes to a very slow increase in the crystal size. The appearance of straight horizontal fringes above the crystal in the infinite fringe setting, and opposed curvature of wedge fringes in the far-field can thus be taken as the limiting point where the growth process is to be terminated.
Figure 5.17: Evolution of schlieren images around the growing crystal from
an aqueous solution. Images have been contrast-enhanced to reveal clearly
the regions of high brightness. In (h) the original photograph as recorded by
the camera is shown. The crystal position has been highlighted.
Module 5: Schlieren and Shadowgraph
Lecture 31: Results and discussion related to crystal growth (part 1)
Figure 5.18: Evolution of shadowgraph images around the growing crystal
from an aqeous solution. Images have been contrast-improved for clarity. A
bright streak of light indicates the separation of the light solution from the
heavy. The streak is seen to move downwards in (d-h), till it stablizes just
around the crystal.
Module 5: Schlieren and Shadowgraph
Lecture 31: Results and discussion related to crystal growth (part 1)
COMPARISON OF THE THREE TECHNIQUES
The knowledge of transport phenomena in the growth of crystals from their aqueous solution in the free convection regime is important for understanding the fundamental mechanisms involved and for fixing process parameters. As opposed to forced convection, the free convection technique has its own importance, for example in protein crystal growth where it is the only choice because of its delicate structure and the ease of managing defects. The creation of a homogeneous concentration field in the boundary-layer adjacent to the solution-crystal interface and uniform distribution of solute in the bulk solution are two major requirements. Refractive index-based optical techniques can be used to examine the nature of convection patterns as a function of time and hence the quality of the grown crystal.
A comparison of the images of the three techniques shows interferograms to be most vivid, since fringes deform and get displaced in relationship to the local velocity field. Thus, they offer the most direct information about concentration distribution as well as the underlying flow field in the solution. Schlieren and shadowgraph images reveal regions of high concentration gradients in the form of heightened brightness, though the former shows greater sensitivity. A review of Equations 1, 3 and 5 shows that interferograms are easy to analyze, schlieren requires integration of the intensity field, (Lecture 29) while shadowgraph requires the solution of a Poisson equation to recover the local concentration.
The three optical techniques under discussion yield images that are integrated values of the concentration field in the direction of propagation of the light beam. Thus, if the spatial extent of the disturbed zone in the solution is small, the information contained in the image is small. In the context of interferometry, the consequence could be the appearance of too few fringes in the infinite fringe setting and small fringe deformation in the wedge fringe setting. In schlieren and shadowgraph, weak disturbances show up as small changes in intensity and hence contrast. The difficulty can be alleviated in schlieren by using large focal length optics so that small deflections are amplified. In shadowgraph, image quality can be improved by moving the screen away from the test cell. Additional difficulties with interferometry are the need for maintaining identical experimental conditions in the crystal growth and the compensation chambers, careful balancing of the test and the reference beams, and limitations arising from fact that quantitative information is localized at the fringes. This discussion shows that configuring the interferometer as the instrument for on-line process control poses the greatest challenge, schlieren and shadowgraph being relatively simpler. Based on the above discussion, schlieren may be considered as an optimum while comparing the ease of analysis with the difficulty of instrumentation.
Module 5: Schlieren and Shadowgraph
Lecture 31: Results and discussion related to crystal growth (part 1)
INFLUENCE OF RAMP RATE AND CRYSTAL ROTATION ON CONVECTION
PATTERNS
The present section discusses the application of the laser schlieren technique to monitoring convection in a crystal growth process from its aqueous solution, when the process parameters are varied. The cooling rate of the solution determines the amount of excess salt available in the solution for deposition on the crystal, and hence the potential difference for driving the convection currents. Ramp rates of 0.05 and have been studied in the present work. These values are smaller than the ramp rate of Section Comparision of Interferometry, Schlieren and Shadowgraph in a Crystal Growth Experiment; consequently the stratification of the solution seen in those experiments was considerably delayed. The rpm of rotational motion fixes the degree of homogenization of the solution and hence, indicates a reduction in the strength of buoyant convection. Crystal rpm of 0 and 15 are studied through experiments. These values have been selected on the basis of their ability to permit growth of crystals of meaningful quality. The effects of ramp rate of the solution, crystal rotation and the size of the growing crystal have been correlated with the growth rate of the crystal. Results of the transient evolution of the convective field in the growth chamber in the form of two-dimensional schlieren images are reported. The images are quantitatively interpreted in terms of concentration contour maps and concentration gradient profiles. In order to bring out the influence of the process parameters, results have been presented in the following sequence: 1. convection currents at a ramp rate of with rotations of 0 and 15 rpm; 2. convection currents at a ramp rate of with rotations of 0 and 15 rpm; and 3. effect of crystal size.
The nature of fluid motion around the crystal gives rise to in the flow and solutal concentration fields. These are relatively thin zones adjacent to the crystal faces where large changes in velocity and concentration take place. Large concentration gradients are revealed in a schlieren image as a brightened region against a darker background. The images discussed in the following sections show that the stable growth regime of crystal growth is accompanied by thin high intensity zones that originate at the crystal surface. Thus, it is clear that fluid motion and transport occur in the bulk of the solution, but are governed by the physical conditions imposed by the crystal. These are 1. a prescription of fluid velocity in terms of crystal rpm, and 2. concentration levels fixed by salt depletion from the aqueous solution.
Module 5: Schlieren and Shadowgraph
Lecture 31: Results and discussion related to crystal growth (part 1)
Figure 5.20: Schlieren images of the transient evolution of the convective
field around a rotating crystal growing form its aqueous solution. (Ramp
rate= , rate of crystal rotation=15 rpm)
Module 5: Schlieren and Shadowgraph
Lecture 31: Results and discussion related to crystal growth (part 1)
Figure 5.19 and Figure 5.20 show the transient evolution of the convective field for experiments when the growing crystal is respectively held stationary in the solution and rotated at a constant speed of 15 rpm. The ramp rate to cool the solution in the two sets of experiments is. Insertion of the seed into its supersaturated solution can lead to an instantaneous temperature difference between them, followed by an initial dissolution of the crystal. This phase of the experiment is not included in the figures. With the passage of time, thermal equilibrium is established, and density differences within the solution are solely due to concentration differences. Adjacent to the crystal, the deposition of solute from the solution to the crystal surfaces results in a change of concentration and the solution goes from supersaturated to the saturated state. In the absence of rotation, the denser solution displaces lighter solution in the vicinity of the crystal and a circulation pattern is set up around the crystal. The fluid motion is largely in the vertical plane. With rotation, a radial pressure gradient creates an independent circulation loop that forms an alternative basis of solute movement. Here, the fluid particles around the crystal move in the radial direction, but conservation of mass ensures that vertical velocities be set-up once again. In the purely buoyancy-driven mode (0 rpm; also called natural convection ), the strength and orientation of the convection currents is determined by the available concentration difference in the solution at any instant of time, and hence the cooling rate. On the other hand, an externally imparted rotation to the growing crystal (called forced convection ) leads to homogenization of the solution, reduction in concentration gradients and hence a reduction in the strength of convection currents. Except for the initial stages of the growth process (where it is diffusion-dominated), these circulation patterns and their interaction form the basis of the transport of solute from the bulk of the solution to the growing crystal surfaces.
Figure 5.19 shows the sequence of convection patterns in the purely buoyancy-driven mode. Growth in the initial stages of the experiments is accompanied by steady, weak convection, during which diffusion effects can be expected to be significant. Thus, for t = 20 hours, a slow growth of the crystal is to be expected. Concentration gradients are primarily localized in the vicinity of the growing crystal. With the passage of time, the size of the crystal increases, and the gradients grow in strength. This result is brought out in the schlieren images as an increase in the light intensity around the crystal. As defined by the bright region, the resulting flow creates a strong plume directly above the growing crystal. Over a longer period of time (20-90 hours), the plume structure remains unchanged. It indicates a stable growth regime for the crystal, where the buoyant plumes are steady and uniform in nature (Figure 5.19(v-vii)). A gradual evolution of the concentration gradients and the associated buoyant plumes ensure a relatively uniform concentration field in the vicinity of the growing crystal, thus leading to symmetric growth of the crystal at the greatest possible rate. As the crystal increases in size, the convection currents grow in strength. Beyond 90 hours, they are seen to become quite vigorous (Figure 5.19(viii)). Correspondingly, time-dependent movement of the plumes was seen in the experiments. This stage is characterized by local changes in the concentration gradients in the vicinity of the growing crystal, followed by a breakdown in the symmetry of the growth process. It is a limit on the time duration up to which a single growth experiment can be carried out in free convection regime, and the consequent limit on the size of the grown crystal.
Module 5: Schlieren and Shadowgraph
Lecture 31: Results and discussion related to crystal growth (part 1)
Figure 5.20 shows that the convection regime is purely forced for short time, and is governed by crystal rotation (< 8 hours). The stirring effect is to be seen by the streaks of light in the image that spread out deep into the solution. Between 8 and 20 hours, the spread becomes narrower, as buoyancy forces re-direct the plume in the vertical direction. For times greater than 45 hours, the plumes show a swirl component, but are vertically directed. The relative importance of rotation and gravity is governed by the ratio of buoyancy and centrifugal forces. The force ratio can be shown to be proportional to the crystal size; hence buoyancy is the guiding force at later times, when the crystal has become large. However, rotation provides a kinematic condition for fluid motion (in the form of a boundary condition), causing the buoyant plumes to become helical, and hence structured.
Figure 5.21 shows the concentration contour maps (normalized between 0 and unity) around the growing crystal for 0 and 15 rpm.
Figure 5.21: Concentration contours around a growing crystal with the
passage of time with and without crystal rotation. The central vertical filled
band in each plot represents the seed holder. Ramp rate=
Module 5: Schlieren and Shadowgraph
Lecture 31: Results and discussion related to crystal growth (part 1)
The growth rate of the crystal is related to concentration gradients, rather than concentration alone. This aspect is explored in Figure 5.22.
Figure 5.22: (a) Variation of normalized concentration gradients near the
three faces of the growing cystal, (b) horizontal growth of the crystal
relative to its maximum size as a function of time, and (c) photographs of
grown crystals with and without crystal rotation. (Ramp rate= )
Module 5: Schlieren and Shadowgraph
Lecture 31: Results and discussion related to crystal growth (part 1)
The variations of the dimensionless concentration gradients averaged over each of the three different faces of the growing crystal (left <001>, right <001> and base <100>) with respect to experimental run time are shown in Figure 5.22(a) for 0 and 15 rpm. As expected, left-right symmetry of the crystal is realized in growth with and without rotation. The effect of rotation is to lower the overall concentration gradient when compared to that generated by buoyancy alone. The gradients on the lower face are small in comparison to the sides. On the lower face, density stratification is stable and buoyant motion is inhibited. The effect of rotation is then to increase the gradients here by inducing fluid movement. Evolutionary profiles on the top face of the crystal are not shown, since the image is adversely influenced by the presence of the seed holder.
In the purely buoyancy-driven experiment, the gradients on the side faces grow in strength with time, whereas the gradients along the lower face are small. The increase in the gradients on the side faces is consistent with the corresponding high intensity regions in the schlieren image sequence shown in Figure 5.19. The problem of high concentration gradients during the later stages of experiments ( t = 60 hours) and also a significant difference in the relative distribution of these gradients over sides and lower faces of the growing crystal is seen to be overcome by rotation. The effect of rotation in equalizing the strength of the gradients over the three faces of the crystal is indicated by the proximity of the gradient profiles in Figure 5.22(a). Figure 5.22(b) shows the horizontal growth of the crystal with respect to the experimental run time. The growth rate with rotation is slightly lower when compared to that based on buoyancy alone; it is however practically linear. The growth rate with crystal rotation is comparatively lower because of two factors: (a) the lowering of concentration gradients in the vicinity of the growing crystal due to homogenization of the solution induced by crystal rotation, and (b) the rotation of the crystal introduces a radial (outward) velocity component that inhibits the transport of solute to its growing surfaces. Figure 5.22(c) shows photographs of the grown crystals for the two cases. The size of the finally grown crystal (after 90 hours) is larger in buoyancy-driven convection, but the crystal quality is superior in terms of transparency when growth is accompanied by rotation.
