DOI: 10.1021/cg9000413
A Computational Method for Extracting Crystallization Growth and Nucleation Rate Data from Hot Stage Microscope Images
2009, Vol. 9 5061–5068
Andrew G. F. Stapley,*,† Chrismono Himawan,† William MacNaughtan,‡ and Timothy J. Foster‡ †
Department of Chemical Engineering, Loughborough University, Loughborough, Leicestershire, LE11 3TU, U.K., and ‡Division of Food Sciences, School of Biosciences, University of Nottingham, Sutton Bonington, LE12 5RD, U.K. Received January 15, 2009; Revised Manuscript Received September 29, 2009 w This paper contains enhanced objects available on the Internet at http://pubs.acs.org/crystal. n
ABSTRACT: A novel algorithm for the computer-based analysis of a sequence of optical microscope images of a crystallization has been developed to extract growth and nucleation data. The algorithm subtracts grayscale pixel values in corresponding positions on successive images, thereby locating pixels relating to new growth which are then either assigned to newly identified (nucleated) crystals or digitally “grown” onto existing crystals. Thus, the algorithm tracks the natural processes of nucleation and growth. The result is a series of maps that identify pixels with specific crystals by an assigned number (label) which remains the same for each crystal from image to image, thereby enabling the growth of any crystal (which can be of any shape) to be tracked. These maps are able to be analyzed (again by computer) to extract unimpinged crystal size and number information, and hence provide crystal growth rate, nucleation rate, and solid fraction data, and crystal size distributions. The method is mainly demonstrated for the isothermal crystallization of tripalmitin from the melt, but an example of isothermal crystallization of sucrose from aqueous solution is also presented.
*Corresponding author. E-mail:
[email protected]; tel: þ44 (0)1509 222525; fax: þ44 (0)1509 223923.
technique in which individual crystals can be monitored as they grow appears to be optical hot stage microscopy (or similar). The well-controlled conditions and ready visualization of nucleation, growth, and morphology for individual crystals over time provide an excellent avenue down which advances in fundamental understanding of crystallization (and recrystallization) processes can occur. The static nature of the system and thin sample geometry used does mean that mass transfer in the liquid phase will be almost purely diffusive rather than convective. This is clearly a very different fluid mechanical environment to that found in most process equipment such as stirred tank crystallizers or scraped surface heat exchangers where large shear forces are typical, convective mass transfer is significant, and growth in three dimensions can occur. This limits the use of optical microscopy results to such systems, but may still be useful for gaining a deeper understanding of some aspects of these processes (for example, of factors that affect shape, which is still relatively poorly understood9). Nevertheless, there are many instances in food and polymer processing whereby crystals do form in an almost static environment, and in these cases optical hot-stage microscopy will be of direct relevance. Optical microscopy has been notably used in the field of fat crystallization to estimate percent solid fraction and extract fractal information,10,11 and to measure growth rates.12 A series of microscope images provide an extremely rich source of data for which to analyze growth rates or nucleation rates. However, investigations of growth that have used optical microscopy have generally employed manual methods to process the data,12,13 but these are labor intensive and timeconsuming. There are obvious advantages if an image analysis program can automatically extract useful information from crystallization images. The objective of this paper is to present a novel algorithm by which a computer program can exploit
r 2009 American Chemical Society
Published on Web 10/29/2009
Introduction The use of imaging techniques is a rapidly growing field in the monitoring and control of crystallization processes. To date, imaging techniques have mainly been applied to the study of stirred tank crystallizers using either in situ or ex situ methods to obtain the images, and increasingly sophisticated image analysis algorithms have been developed to analyze these images to produce both size and shape data.1-7 For nonspherical crystals, more than one size dimension can be extracted which leads to a more complete characterization of the crystal dimensions than is possible from laser-based sizing and counting techniques (such as laser backscattering and laser diffraction) which are restricted to reporting a single particle size (or chord length) per crystal. The two-dimensional (2D) particle size maps that can be created from imaging methods have particular uses for characterizing crystals which have a high aspect ratio,5 and it also allows for the possibility of polymorph determination without relying on additional methods such as X-ray diffraction or infrared spectroscopy.2,8 Image analysis is not without its problems however, chief being the segmentation of images (removing background noise to reveal only the crystals), the problems associated with deconvoluting size and shape information of three-dimensional (3D) objects from 2D images, and the large computational burden. As well as using image analysis for crystallizer control and monitoring, the data can be used in conjunction with models to infer nucleation and growth rate information.1,7 However, it is not yet possible to track the whole growth history of an individual crystal in stirred tank crystallizers. The only current
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Figure 1. Source and processed images from the crystallization of tripalmitin at 49 °C, (a) image after 270 s hold time, (b) image after 290 s hold time, (c) movement image (panel a subtracted from panel b), (d) binary image showing new pixels (after applying a threshold to panel c). The width of each image corresponds to 1.175 mm.
the rich source of data that a series of images present, and extract meaningful information in an automated manner. This is demonstrated here for crystallization of the tripalmitin from the melt (tripalmitoylglycerol) and sucrose from aqueous solution, but the method itself should be applicable to the crystallization (or any analogous processes) for almost any system that can be imaged in this way. Experimental Section Tripalmitin (>99% pure) was obtained from the Sigma Chemical Co. (Poole, UK) and sucrose (analytical grade) was obtained from Fisher Scientific (Loughborough, UK). Both were used without further purification. An aqueous solution of sucrose (75% sucrose) was made by weighing sucrose and distilled water into a beaker and gently stirring using a glass rod on a hot plate at 90 °C. Distilled water was added at intervals to compensate for the evaporation of water and maintain the overall sample mass. Optical microscopy was carried out using a Linkam THMS600 variable temperature stage (Linkam Instruments, Tadworth, UK) with a Leitz (Diaplan) microscope coupled to a PixeLink PL-A662 digital camera (PixeLink, Ottawa, Canada) and Linksys 32 software data capture system (Linkam Instruments, Tadworth, UK). For tripalmitin experiments, a small sample (∼1 mg) of solid tripalmitin was weighed on a circular glass slide which was then preheated at 90 °C to melt the sample. Another coverslip was then placed concentrically on top of the drop to ensure a uniform thickness of the sample. The sample was subsequently placed in the temperature stage, where it was heated to 90 °C and held for 5 min to obtain
an isotropic melt. Thereafter, the sample was cooled at a rate of 50 °C min-1 to an isothermal holding temperature. Fast cooling was achieved by flowing liquid nitrogen through the stage. Images (1280 1024 pixels) were collected during the isothermal holding period, observed via a 10 objective lens and captured automatically every 5-30 s (depending on the temperature and holding period) using Linksys software. The images were scaled by taking an image of a graticule at the same magnification. A limitation of the experiments was that as high solid fractions were reached voids would appear on the images marked by a strong meniscus. This is a consequence of the higher density of the solid phase which thus takes up a smaller volume than the parent liquid phase. As the separation of the glass slides is essentially fixed from when the first crystal growth occurs, the volume reduction is accompanied by flows of liquid into the image area from the rest of the sample. However, once a section of fluid becomes isolated from this means of supply, by becoming completely surrounded by solid crystals, it then it is susceptible to forming voids. The formation of these voids obviously disrupts the crystallization process where they meet a growing crystal interface, and so the isothermal hold was stopped at this point. Samples were reused for further experiments after heating at 1 °C min-1 to ∼95 °C (which allowed melting to be visualized) and holding for 3 min before once again cooling to another isothermal temperature. Experiments with sucrose were performed in a similar manner, except that during loading the slide was not preheated and the stage was held at room temperature. As soon as the coverslip was placed on the sample in the stage, the stage temperature was raised to 50 °C. As nucleation rates are very low for this system, it is unlikely to see nucleation events occurring in a single field of view. Hence a strategy
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Figure 2. Flowchart showing the outline of the algorithm used to produce the crystal-maps. It is assumed that all images are loaded as grayscale images. adapted from Human et al.13 (in their study of naphthalene crystallization from toluene) was used by which the whole sample (not just a single field of view) was scanned for the appearance of crystals. As soon as a growing crystal was found, the temperature was raised to reduce the size of the crystal. Once dissolution of the crystal was observed to begin, the temperature was then only raised very slowly and carefully (in steps of 0.2 °C approximately once per minute) until the crystal was barely visible. The stage was then held at this temperature (92.5 °C) for 10 min to allow concentrations in the solution to equilibrate further (again taking care that the crystal did not dissolve completely), before cooling at 20 °C min-1 to 50 °C and holding isothermally for approximately 100 min. The image analysis was performed from the first image in which the isothermal temperature of 50 °C was reached, by which time the crystal had grown slightly in size.
Image Analysis Method One of the chief difficulties for image analysis is segmentation, that is the identification of meaningful objects (i.e., crystals) and the removal of unwanted artifacts such as background dust particles in the optics and other noise from the images. These artifacts are easily filtered out by humans (who are equipped with superb methods of pattern recognition) but are difficult to remove via a computer algorithm if a single image is analyzed in isolation. However, this problem can be overcome by “subtracting” successive images from each other, whereby the grayscale value corresponding to each pixel position in an image is subtracted from the grayscale value of the corresponding pixel position in the following image to produce a “difference” image. For the vast majority of pixels, the grayscale value will not have changed significantly and thus the difference in image pixel values is either zero or close to zero. However, where “movement” occurs between the images then the subtracted values in affected regions are significantly different from zero. “Movement”
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can result from the sample moving or a particle or bubble moving through the field of view, but in the vast majority of cases this will be the result of crystal growth. This is illustrated in Figure 1, which shows two successive images during the crystallization of tripalmitin at 49 °C, and the difference between them. The image shown in Figure 1b was taken 20 s after that of Figure 1a, and thus the crystals in Figure 1b are slightly larger than in Figure 1a. When Figure 1a is subtracted from Figure 1b, the result (Figure 1c) shows bright rings where growth has occurred. This can then be thresholded to produce a binary (black/white) map of “new” pixels (Figure 1d). These binary maps thus show the appearance of newly crystallized material. These can then be used to track crystallization through a time sequence of images, whereby a sequence of “crystal-maps” corresponding to each image is constructed. By “crystal-map”, we mean an image (a 2D array) that is derived from a microscope image in which all the pixels associated with a specific crystal are all labeled with the same integer value. The crystal-maps can be visualized by representing each number by a different color. An example crystal-map is shown (in grayscale for the print version of this article) in Figure 3a. However, not only does this allow each crystal to be identified visually, but it also enables further analysis to be made by a computer program to analyze the size of each crystal and count the total number of crystals on each image. The crystal-maps for different images are also linked in that each crystal retains the same number label in all images in which it appears, so that crystal growth can be tracked from image to image. The algorithm to derive these crystal-maps is presented schematically in Figure 2. The program starts with images at the very beginning of a crystallization experiment, just before any crystals have appeared. The initial condition is thus a blank crystal-map, that is, all pixel values are zero. The program steps through successive images in turn, operating in two modes: “nucleation mode” and “growth mode” to mirror the crystallization process. For any given image the program first runs in “nucleation mode” in which the sole aim is to detect any newly formed crystals from the current binary image of “new pixels”. The criterion for a new crystal is when an object (defined as any connected group of new pixels) is identified and is found to be (i) within a certain size range (based on the number of pixels in the object) and (ii) does not overlap with or come very close to an existing crystal. The size range is chosen to minimize the chance of false crystals being identified and to minimize the chance that a genuine crystal appearance being missed. When such an object is found all the pixels in that object are loaded into the crystal-map with an integer label which is unique to that object. When all objects are examined the nucleation step for that image is complete. The program then switches into “growth mode” where remaining new pixels that are near or next to existing crystals are “grown” onto the nearest crystal by assigning them the same integer label as that crystal. Only pixels that are within a predefined distance from a crystal can be grown onto a crystal in this way. The maximum allowable distance is set by the user to be the expected maximum growth distance of any crystal between images plus a margin for error. Figure 3 shows the new pixels from Figure 1d have been used to update the crystal-map corresponding to Figure 1a (Figure 3a) to that corresponding to Figure 1b (Figure 3b). The twin steps of nucleation and growth are then applied in turn to successive images, thus building up the series of
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Figure 3. Crystal-maps corresponding to source images (a) Figure 1a, and (b) Figure 1b. The width of each image corresponds to 1.175 mm.
Figure 4. (a) Source image and (b) corresponding crystal-map for the crystallization of tripalmitin at 49 °C after 540 s hold time. The width of each image corresponds to 1.175 mm.
crystal-maps over the whole time course of the crystallization (see accompanying videos 1, 2, 3, 4, 5, and 6 in avi format). An example crystal-map corresponding to an image later in the crystallization (Figure 4a at time 540 s) is shown in Figure 4b. It can be seen that individual crystals that would be very difficult to discriminate in this image are identifiable, by virtue of the fact that their growth has been followed over time. Some small imperfections in the maps are noticeable in Figure 4b. These are the back rectangle in the top left-hand corner, the black spot on the bottom edge, and the black triangles in the top right and bottom right corners. These can all be seen to originate from black areas on the original images (which are black on each image) and is therefore a deficiency with the experimental setup rather than the image analysis method. Variations to the method can be made to increase robustness. For example, image subtractions can be made with images that do not immediately follow one another (say two apart) to reduce the chance of growth pixels being missed in the thresholding process. The nucleation spotting algorithm can also include a further check to ensure that the nucleated crystal continues to grow in subsequent steps. Both these methods were used here. The benefit of this overall approach (which we believe to be new) is that it now enables a vast amount information to be automatically extracted from these crystal-maps as follows: (1) Although it is also possible by simply thresholding each image, the solid fraction for an image can be calculated from the crystal-maps by adding all nonzero pixels and dividing by the total number of pixels. This assumes that a nonzero pixel represents 100% solid and a zero pixel represents 100% liquid (which may not be completely accurate when structural elements are present whose dimensions are of a similar size or smaller than the pixel dimension). It is also assumed that all the crystals extend the full height between the glass slide and coverslip and that the cross section area does not vary with
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height. Given these assumptions, the time course of solid fraction (and hence liquid fraction or solid/liquid ratio if desired) can be followed by extracting this information from successive images. (2) The number of crystals can be tracked versus time. This can give a measure of nucleation rate per unit volume if the number is divided by the observed sample volume. Crystals are not counted if they were first observed at the periphery of the image (within 10 pixels of the edge) as these are likely to have first nucleated outside of the field of view. Thus, the sample volume used to calculate the nucleation rate is obtained by calculating the main area of the image (not including pixels within 10 pixels of the edge) multiplied by the thickness of the sample. Of course, an additional factor to consider is that the precise moment of nucleation is never seen as the nucleus is too small to be observed. The technique allows the possibility of extrapolating the growth curve of a crystal (see point 4 below) back to zero size to estimate an effective nucleation time for each crystal. However, this was not performed here as the time adjustments for each crystal were small and not significantly different between crystals to have a noticeable effect on the shape of the crystal number versus time plot. (3) However, it is found that in general, the nucleation rate per unit volume of sample diminishes as time proceeds. The main reason for this is that nucleation can only occur in the liquid phase and the volume of the liquid phase decreases over time (solid/solid transformations are not considered here). A nucleation rate per unit volume of liquid (n) can be defined as follows: n ¼
1 dN VL dt
ð1Þ
where VL is the volume of liquid, N is the number of crystals, and t is time. Integrating with respect to time (assuming, for now, a constant nucleation rate) with an initial condition of zero nuclei at the end of the induction time period gives Z t Z t Z t VL n dt ¼ n VL dt ¼ nVT XL dt ð2Þ N¼ tind
tind
tind
where VT is the total volume and XL (= VL/VT) is the liquid fraction. This gives rise to a modified time variable (tmod) defined: Z t tmod ¼ XL dt ð3Þ tind
So plotting a graph of N versus tmod should give a straight line graph of gradient nVT for a constant nucleation rate. The modified time is effectively a clock that ticks at a rate that is proportional to the fraction of liquid present, as dtmod ¼ XL dt
ð4Þ
When completely liquid the modified time advances at the same rate as the actual time, but as XL diminishes the rate of “ticking” decreases. If the nucleation rate is not constant then the above plot is still useful as the value of nVT at any point can now simply be extracted from the instantaneous gradient of N vs tmod. This
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can be expressed mathematically by combining eqs 1 and 4. Thus, the variation of n with t can be plotted. n¼ ¼
1 dN 1 dN VL 1 dN ¼ XL ¼ VL dt VL dtmod VT VL dtmod 1 dN VT dtmod
ð5Þ
(4) Individual crystals can be analyzed to show the growth of a front over time and hence extract growth rates. The precise method will depend on the shape of the crystals. Circular spherulites (as observed with tripalmitin) can be analyzed to produce radius versus time data. The unimpinged radius of a crystal can be estimated from the average radial distance (from where the crystal first nucleated) of pixels added in the last growth step. This method was chosen for its robustness as it is not strongly influenced by outlier pixels. This measurement allows the radial growth of each crystal to be followed with time, and enables growth rates to be evaluated. Crystals that have nucleated off screen are likely to produce false results by this method as the visible segment of a large front will appear to grow much more rapidly (within the confines of the image) than the actual frontal growth rate as it crosses into the field of view. Hence, crystals that are first seen at the periphery of the image (within 10 pixels) are excluded from growth rate analyses. The growth of nonspherical crystals (such as polygons) can also be followed but requires an algorithm to identify the directions along which growth is to be measured. This was performed in the case of sucrose which produced a crystal in the shape of a parallelogram. (5) A crystal size distribution can be evaluated using either the size data gathered in (4) or based on equivalent area; that is, for a crystal an equivalent diameter can be defined as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 area ð6Þ i:e:; d ¼ π This latter definition might be regarded as a more useful method for characterizing crystal size when significant impingement occurs, as it is directly related to the quantity of matter in each crystal. (Conversely, such a definition is not helpful for assessing crystal growth rates which are best served by measuring unimpinged radii as described in (4) above). The data can be plotted either as a cumulative distribution or a number density distribution, but unless the total number of crystals is large (such that a reasonably large number of crystals are allocated to individual bins) the number density plots are unlikely to show meaningful information. These problems could be overcome by repeating the experiment a number of times and merging the information to form a larger data set. There are also some inherent inaccuracies with the method as crystals that cross the edge of the image cannot be viewed in their entirety and so their area cannot be accurately measured. One strategy could be to discount any crystals overlapping the edge, but this would unfairly penalize large crystals which have a greater chance of doing this. The method used here was to only include crystals that had nucleated on-screen, that is, those that are counted as nucleation events in (2) above, and estimate the size of any crystals that currently overlap the boundary (i.e., have pixels at the edge of the screen) by measuring the half of the crystal area that is fully observed (those pixels further away from
Figure 5. (a) Source image and (b) corresponding crystal-map for the crystallization of tripalmitin at 47 °C after 150 s hold time. The width of each image corresponds to 1.175 mm.
the edge than the center or nucleation point of the crystal) and multiplying by two. Crystals at the corners of the images which cross two sides would have their area based upon the quadrant lying furthest from both edges (that which is fully observed) and multiplying by four. (6) Induction times can also be extracted, although this is easy to do manually. Ideally, an estimate of the true nucleation time point of the first crystal is estimated from extrapolating its growth curve back to zero size. All processing was performed in MATLAB (The Mathworks Inc., Natick, MA) using routines from the Image Processing Toolbox. The analysis was performed by two separate programs. The first program creates the crystal-maps from the raw images, and the second program extracts the crystallization data from the crystal-maps. Results and Discussion The results from the tripalmitin experiments are presented first. The identification of polymorphs was performed by examining remelting behavior and by comparison with the previous studies of isothermal tripalmitin crystallization by microscope12 and differential scanning calorimetry (DSC).14 Under the isothermal conditions studied here however, only the β0 polymorph (melting point 55-57 °C12) and not β was observed to directly crystallize. The primary evidence for this was that all crystals in all experiments could be seen to transform into another polymorph almost immediately on remelting. This occurred from the crystal centers in the same way as described by Kellens et al.12 We were not able to directly crystallize the β form of tripalmitin, even after waiting for many hours at temperatures which had produced the β form in DSC experiments after only a few minutes. Kellens et al.12 were able to crystallize the β form but only by briefly nucleating a lower polymorph at a lower temperature, raising the temperature to transform to nuclei into the β form and then allowing it to grow. The difference between the behavior observed by DSC and microscope is presumably due to the sandwiching of the sample between two microscope slides which results in a very thin sample. This may simply reduce the likelihood of β nucleation (as a result of the very small sample volume) or it may be a consequence of a virtually zero shear environment as even natural convection currents are not present (which may occur in a DSC pan). The formation of the β form is widely believed to be influenced by shear or fluid movement for reasons that are still not entirely understood,15-17 and it may be the case that even a small amount of fluid movement that may occur in a DSC is important in the formation of the β form.
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Figure 6. Evolution of solid fraction for crystallization of tripalmitin at various temperatures extracted from microscope images.
Figure 7. Unimpinged crystal radii extracted from microscope images of tripalmitin crystallization at 49 °C.
Figure 9. Number of crystals plotted (a) versus time and (b) versus modified time as defined in eq 3 (data are for tripalmitin at 49 °C).
Figure 8. Extracted radial growth rate of tripalmitin versus temperature.
Figure 10. Nucleation rate of tripalmitin per unit volume of liquid versus temperature.
The morphology of the crystals was rougher and more diffuse at lower isothermal hold temperatures, and more perfectly circular at higher temperatures as found in earlier studies18 (see Figure 5 and accompanying videos 1, 2, 3, 4, 5, and 6 in avi format). Nevertheless, the algorithm for producing the crystal-maps was robust enough to cope with the variable crystal morphology. Calculated values for solid fraction versus time are shown for various temperatures in Figure 6. Naturally, the faster
transition is found at lower temperatures. These data can be compared to DSC14 which shows broadly similar behavior for the two temperatures which can be directly compared (48.5 and 50.5 °C). An example plot of unimpinged radius versus time is shown for 49 °C in Figure 7. The striking feature of this plot is that (with very few exceptions) the growth rate is virtually constant over all crystals and all times. Thus, for this system growth rate can be considered to be independent of crystal radius.
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Figure 11. (a) Evolution of crystal size number with time (increasing with arrow) for tripalmitin at 47 °C at 20 s intervals. (b) Crystal size distributions at a solid fraction of 30% versus temperature.
Figure 12. (a) Source image and (b) corresponding crystal-map for the crystallization of sucrose from 75 wt % sucrose in water solution at 50 °C after 97 min hold time. The width of each image corresponds to 0.843 mm.
Such a result would be difficult to demonstrate by manual methods in such a convincing manner. An average growth rate for each temperature can thus be extracted by fitting a gradient to these lines. These are presented in Figure 8, and compare well with literature data.12 The number of crystals at 49 °C is plotted against time in Figure 9a, which shows a tailing off of the nucleation rate above 300 s. However, the plot of crystal number against modified time (tmod) (Figure 9b) is much more linear although the curve is somewhat “jagged”. Given that nucleation is a
stochastic process and only small numbers of crystals are counted this variation is not unreasonable, and thus one can conclude that nucleation rates per unit volume of liquid can also (like growth rate) be considered to be a constant for a particular temperature. The nucleation rate per unit volume of liquid can be extracted from a linear fit and is plotted against temperature in Figure 10, where it is fitted to a simple exponential variation with subcooling below the liquidus. The variation of nucleation rate with temperature was very much larger than for growth, typically showing 2 orders of magnitude variation over only a 3 °C range. As described earlier, the evolution of the crystal number distribution can be plotted, and this is shown for 47 °C in Figure 11a (based on a bin size of 5 μm). Cumulative crystal size distributions for different temperatures can also be compared and this is done for a constant value of solid fraction of 30% as shown in Figure 11b. The trend of much smaller crystal sizes produced at lower temperatures is obvious from the images themselves, and is a reflection of the much larger temperature variation of nucleation rate than crystal growth. An example image and crystal-map from the crystallization of sucrose from a 75 wt % sucrose solution is shown in Figure 12. The map only shows one crystal but illustrates the ability of the technique to handle crystals of other shapes (such as parallelogram). A growth analysis was done by
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Figure 13. Growth curves of the four faces of the sucrose crystal shown in Figure 12.
manually identifying four lines (coincidentally aligned with the vertical and horizontal directions of the image) along which the frontal positions were plotted. The results of this are shown in Figure 13. A curious feature is the two speed growth of some of the faces, which has been highlighted by this analysis method. This can also be noticed by playing back the accompanying video (4 in avi format). The precise reasons for this can only be speculated upon by the authors (the faces may be temporarily “poisoned” or otherwise disrupted) and merits further investigation. Another surprising feature of the curves is that (with the exception of the step changes) the growth rates do not appear to diminish with time, as one might expect the sucrose supersaturation to fall as sucrose is depleted from the liquid phase. However, it should be pointed out that the observed crystal was the only crystal growing on the microscope slide (this was checked afterwards) and as such the volume of the mother liquor was at least 2 orders of magnitude greater than that reached by the crystal at the end of the experiment. So the average concentration of sucrose in the mother liquor was unlikely to be significantly affected over the course of the experiment. The crystal can therefore be considered to be growing in an “infinite sea”. Whether the crystal growth can be considered to be diffusion controlled or reaction controlled can be judged by comparing the crystallization and dissolution behavior (dissolution was performed in the preliminary stages of the experiment to reduce the size of the crystal to the smallest size possible without causing complete dissolution). It was found that dissolution was much more rapid than growth. This is a commonly observed phenomenon, and this is generally attributed to the melting rate being controlled by diffusion only, whereas growth is controlled by an additional surface reaction step.19 Further evidence for this is that while during crystal growth an almost perfect parallelogram crystal shape was maintained, during the previous dissolution step the corners were steadily “eroded” and the crystal becomes rounded - with the corners dissolving preferentially because of the closer proximity to lower sucrose concentrations. If growth were controlled by diffusion then the corners would grow preferentially. The parallelogram shape of the crystal is established and maintained from the relatively slow rate at which new crystal layers nucleate compared with the rate at which existing layers are completed. The lack of diffusion control during growth can be supported by “back of the envelope” style calculations for diffusion distances of water. The self-diffusion coefficients of water and sucrose in aqueous sucrose solutions at 50 °C and 74 wt %
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sucrose have been measured by pulsed gradient NMR as 1.15 10-10 m2 s-1 and 5.8910-12 m2 s-1, respectively.20 As water is the much faster diffuser, most of the diffusion occurring will be as a result of water diffusion (rather than sucrose diffusion). Nominal diffusion distances21 (Dt)1/2 of water over time scales of 1, 10, and 100 min can thus be calculated as 83, 262, and 830 μm, respectively, which are an order of magnitude greater than the actual crystal growth rate, although perhaps not fast enough to completely flatten out the concentration profiles in each phase. Note that we do not believe natural convection plays a role in these experiments (at least during the isothermal periods) due to the low temperature gradients in the sample and the thinness of the samples. Indeed, while fluid motion is observed (from the motion of tiny specks of particulate material) during fast heating and cooling of the stages, this motion is always observed to stop shortly after isothermal conditions are reached. In summary, the use of image analysis in combination with hot stage microscopy is a useful complementary technique that can provide fundamental information about crystallization which might be difficult to elucidate by any other technique. Acknowledgment. We thank the Biotechnology and Biological Sciences Research Council for funding this work (grant reference D20450).
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