Evaluation of Size and Shape of Crystalline and Amorphous Particles

Article pubs.acs.org/crystal

Evaluation of Size and Shape of Crystalline and Amorphous Particles from Diffraction Pattern Krzysztof Pieszyński* Institute of Physics, Lodz University of Technology, ul. Wólczańska 219, 93-005 Łódź, Poland S Supporting Information *

ABSTRACT: In this Article, the model of experimental setup enabling observation of the early stages of growth of single crystalline and amorphous particles from a solution is examined. Based on the scalar theory of light, it is shown that by using the diffraction pattern of such particles, both characteristic size and position of a few particles of micrometer-sized, even considerably distant from each other, can be simultaneously measured. Moreover, solution volume in which the investigated process takes place is relatively large, so that process can occur undisturbed.

1. INTRODUCTION Very frequently, there is a need for investigation of crystalline or amorphous particles in the early stages of their growth. A specific example might be research on the formation of urinary stones.1 Research of this type often requires the direct observation and measurement of various parameters during particles growth, in particular, under in vitro condition. Evaluation of sizes and shapes of crystalline or amorphous particles during their growth seems to be especially relevant. For the measurement of the above-mentioned parameters, widely used techniques are based on optics. A characteristic feature of such techniques is the fact that, generally speaking, a light beam does not affect the process of a particle’s growth. An example would be, inter alia, techniques based on the properties of the scattered light2−4 or spectrophotometry.5 Many of these techniques do not make estimates individually, but rather statistically such values of the particle’s parameters as their size or shape since a probe light beam illuminates in the same time a large number of crystals. Undoubtedly, the most accurate way of measuring size and shape of individual particles in the early stages of their growth is provided by an appropriately selected type of microscope. It may be, for example, an optical microscope. The problem here is that such a measurement requires either collecting particles for the measurement from the places where they grow, which interferes with the growth process itself, or the particle’s growth must take place directly in the field of view of the microscope. In the latter case, the sample volume is often so small (particularly when the observation is carried out under high magnification) that it may have an impact on the process of the particle’s growth. However, even when some special type of microscope can work with a relatively large sample size, it is difficult to make a simultaneous observation of a few particles located at various depths because optical microscopes in principle provide high-quality images in the single plane perpendicular to the optical axis. Simply, the depth of field shrinks with increased magnification. © 2015 American Chemical Society

It seems that one way of overcoming the last of the described problems is the use of the diffraction pattern produced by the investigated particles. Undoubtedly, the classic Fraunhofer’s diffraction pattern of a crystalline or amorphous particle is suitable for the evaluation of both size and shape of such a particle6 and does not require reducing the size of the cuvette, in which the investigated growth takes place. However, the problem arises when there are several particles in the examined area. It is well-known that the classical Fraunhofer’s diffraction pattern is invariant under spacial translation of the objects which are the source of the diffraction. This fact means that such a diffraction pattern cannot localize the position of these objects. Thus, the readout of detailed information on parameters of the individual particle from Fraunhofer’s diffraction pattern can be very difficult. It turns out that the insertion into the light beam of a relevant optical system allows identification of the sets of interference fringes (in the entire diffraction pattern) which are produced by the individual particles, so that the interpretation of such a diffraction pattern is much easier. There are a lot of experimental setups operating in accordance with the rule described above. This Article analyzes the usefulness of this type of experimental setup for evaluation of the size and shape of crystalline or amorphous particles in the early stage of their growth process. The analysis is based on a single example of such a setup.

2. MODEL OF EXPERIMENTAL SETUP Figure1 presents the model of the experimental setup which is discussed here. The model assumes that a water drop of diameter d = 2 mm is illuminated by a Gaussian beam with parameters which are readily achievable by the light beam emitted by commercially available He−Ne lasers, i.e., the wavelength Received: November 20, 2014 Revised: May 3, 2015 Published: June 1, 2015 3137

DOI: 10.1021/cg501696y Cryst. Growth Des. 2015, 15, 3137−3143

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Crystal Growth & Design

Figure 1. Model of the experimental setup which is discussed in this paper. A water drop is illuminated by a Gaussian light beam. The waist of the Gaussian beam is located at a distance of z1 (plane P1) before the water drop. Inside the water drop, at a distance of z2 (plane P3) behind the drop surface (plane P2) is an obstacle which represents the small crystalline or amorphous particle. Diffraction pattern of the obstacle is produced in far field (plane P5).

λ = 632.8 nm and the beam waist w0 = 0.64 mm. The intensity of light on the axis of the beam at its waist is set to unit value. The waist of the Gaussian beam is located at a distance of z1 = 0.5 m (plane P1) before a water drop. Inside the water drop, at a distance of z2 (plane P3) behind the drop surface (plane P2) through which the light beam enters the water drop, is an obstacle which represents the small crystalline or amorphous particle in the early stage of its growth. The diffraction pattern of the obstacle in a far field (P5) is analyzed. This diffraction pattern is computed using the method of Fourier optics,7 which is based on the scalar diffraction theory. This fact, of course, introduces a long list of presuppositions limiting the generality of the presented method. The scalar theory assumes8 that light is propagating in a dielectric medium which is homogeneous, isotropic, linear, nonmagnetic, and nondispersive. All of these conditions are satisfied with a good approximation for both water and air being the media in presented model. For example, the water dispersion can be ignored since the monochromatic light is used in the calculations. However, it should be taken into account that in a real experiment a variety of solvents are used rather than water, so in some cases, the proposed method of evaluating the parameters of single crystals or amorphous particles may be subject to significant measurement errors due to inaccurate fulfillment of above conditions. In addition, it is assumed that the spherical aberration associated with the passing of light through interfaces between water and air can be ignored. This last assumption is not fully met only for particles located very close of the edge of Gaussian beam. Finally, the flat obstacles themselves used in the presented model are a certain approximation of real objects; however, it seems that in most cases it is a sufficient approximation and in many cases the discussed experimental setup at least makes it possible to observe the changes in examined parameters. Chemical compounds, whose molecules due to the long-range structural order or their optical activity strongly influence the light polarization, clearly do not meet the conditions of scalar diffraction theory. Liquid crystals may be mentioned as an example of such compounds. Such compounds affect the polarization of light and therefore do not meet the scalar theory of light. The diffraction patterns obtained after passing through such substances may differ from those described herein. The computations are performed as follows. Using the parameters of the Gaussian beam in the plane P1, which are

specified in the previous paragraph, the electric field in the plane P2 is computed. This is done on the basis of the well-known properties of the Gaussian beam.9 Then, it is assumed that in the plane P2 is an ideal thin lens of a focal length which is the equivalent to focal length associated with the curved air−water boundary,10 and the electric field is computed just behind this lens, that is to say, after passage of light through the air−water boundary. For this purpose the transmittance function for the fine focus of an ideal thin lens is used.11 Subsequently, the electric field in the plane P3 is computed. In this plane, obstacles that simulate crystalline and amorphous particles are located. The objects that perfectly absorb the light are chosen as obstacles. Herein, the obstacles in the form of flat discs represent amorphous particles and the obstacles in the form of flat rectangles represent single crystals. Therefore, for computation of the electric field in the plane P4, the field in plane P3 is used; however, in the places where the obstacles are placed, the electric field is assumed to be zero. Computations of the electric field on the exit surface of the water drop are carried out similarly to those on the entrance surface. So, the electric field in the light beam is calculated on the plane P4 just after the exit of the water drop. All computations within the drop are performed using the Rayleigh-Sommerfeld transfer function propagator.8,12 However, it is not always possible to meet the critical sampling condition13 for the required sample interval. Therefore, in cases when the distance z2 is less than 1.5 mm, calculations of the electric field in plane P3 are performed using the Rayleigh-Sommerfeld impulse response propagator.8,12 For the same reason, in cases when the distance z3 is less than 1.5 mm, calculations of the electric field in plane P4 are also performed using the Rayleigh-Sommerfeld impulse response propagator. The results obtained using these two methods turned out to be very similar, which is interpreted as evidence of the correctness of these computations. By contrast, the electric field in the plane of P5 (P5 represents infinity) is calculated using the Fraunhofer approximation.8,12 The light intensity in the diffraction pattern observed in the plane P5 (it is shown in most of figures) is computed as modulus squared of the electric field. Figure 2a shows the distribution of the light intensity just behind the three obstacles which are located in plane P3. The three obstacles are numbered as follows: 1, 2 μm × 7 μm 3138


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Figure 2. (a) Positions of three obstacles in the plane P3 (at z2 = 1 mm). The obstacle marked 1 (size 2 μm × 7 μm) represents a single crystal. The obstacles marked 2 (with radius 7 μm) and 3 (with radius 2 μm) represent the amorphous particles. (b) Diffraction pattern from these obstacles in plane P5. Both planes P3 and P5 are defined in Figure 1. Qualitative analysis of the diffraction pattern shows three interesting properties. Images from individual obstacles are spatially separated. Modulation of the intensity of the interference fringes produced by the obstacle 1 is different in horizontal and vertical directions which coincides with its shape. Modulation of the intensity of the interference fringes coming from obstacles 2 and 3 is associated only with their size, i.e., the intensity of the interference fringes coming from obstacle 2 (of larger size) is modulated with a higher spatial frequency than the intensity of a smaller obstacle 3.

Figure 3. (a) Total profile of the intensity of light in the diffraction pattern versus diffraction angle due to obstacle of radius 5 μm which is placed on the optical axis of a Gaussian beam. (b) Net profile of the intensity of light in the diffraction pattern for the same obstacle and in the same position. The arrows indicate three successive orders of suppression of interference fringes.

rectangle (represents single crystal); 2, disc of radius 7 μm (represents amorphous particle), 3, disc of radius 2 μm (represents another amorphous particle), respectively. Figure 2b shows the diffraction patterns of these obstacles in infinity. Figure 2b also demonstrates that different sets of interference fringes can be easily assigned to the relevant obstacles. The diffraction pattern in this figure shows that each of obstacles produces the concentric set of interference fringes. However, the intensity of the individual interference fringe associated with the round obstacles (2 and 3 in Figure 2b) does not depend on the direction in the XY plane, while the intensity of the individual interference fringe

associated with the rectangular obstacle (1 in Figure 2b) clearly depends on the direction. For the system of interference fringes related to a rectangular obstacle along the x-axis (the direction of the longer side of the obstacle 1), the periodic disappearance (suppression) of the interference fringes is seen wherein the “spatial frequency” of the occurrence of this suppression is greater than along the y-axis (the direction of the shorter side of the obstacle 1). Comparison of the set of interference fringes due to the round obstacles shows that, as mentioned above, the intensity of the fringes does not exhibit any directionality in the XY plane and the fringes associated with greater obstacle 2 3139


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Figure 4. Profiles of the net intensity of interference fringes from the disc-shaped obstacles versus diffraction angle. The insets (a), (b), (c), and (d) show the profile for the obstacle’s radius of 2 μm, 3.5 μm, 5 μm, and 7.5 μm, respectively. All the insets concern a situation in which the obstacle is located in the middle of the water drop, on the optical axis of a Gaussian beam.

Figure 5. Profiles of the net intensity of interference fringes due to the disc-shaped obstacle versus diffraction angle. The insets (a), (b), and (c) depict the profile for the obstacle’s positions z2 = 0 mm, z2 = 1 mm, and z2 = 2 mm, respectively. All the insets concern the situation in which the obstacle is on optical axis of the Gaussian beam and has a fixed radius of 5 μm.

Figure 3a shows a profile of the intensity of light in the diffraction pattern of the disc-shaped obstacle of radius 5 μm which is placed on the optical axis of a Gaussian beam. The shape of the profile obtained in this way is more discernible if the intensity of the ’pure’ Gaussian beam is subtracted from the total intensity of the diffraction pattern. The net intensity profile of the diffraction

are suppressed more frequently (in radial direction) relative to the fringes due to the lesser obstacle 1.

3. RESULTS AND DISCUSSION All computations that are presented from now on are done for single obstacle which is placed in the middle of the Gaussian beam. 3140


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Figure 6. Locations of the suppression of the interference fringes versus the radius of the disc-shaped obstacle. The points on each line correspond to the different position of the obstacles inside the drop (on optical axis of Gaussian beam). The highest situated bold line corresponds to the position z2 = 2 mm and the lowest situated dotted line corresponds to the position z2 = 0 mm. Graphs between these lines correspond to the intermediate positions of the obstacles. All points are associated with the first-order suppressions.

Figure 7. Angular position of the diffraction maxima (a) as a function of the obstacle’s position (z2) inside the water drop (b) as a function of the obstacle’s size. The circle indicates a first-order diffraction maxima while a plus sign indicates a second-order diffraction maxima.

pattern, that is to say obtained in this way, is shown in Figure 3b. It is worth noticing that the profile demonstrates two features of this diffraction pattern: first, the characteristic maxima and minima of the interference fringes and, second, repeating suppression (disappearance) of these interference fringes. Three successive orders of these suppressions are indicated by the arrows in Figure 3b.

Due to the symmetry of the diffraction pattern, only one-half of the profile is shown in the subsequent figures. Figure 4 shows the profile of the net intensity of interference fringes from diskshaped obstacles with different radii. During the calculations, it is assumed that all the obstacles are placed in the middle of the water drop (z2 = 1 mm). The analysis of the graphs presented in 3141


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Figure 8. Diffraction pattern from obstacle in the form of a 2 μm × 6 μm rectangle. The obstacle is located in the middle of a water drop, on optical axis of Gaussian beam, and its longer edge is aligned horizontally. (a) “Full” light intensity of the diffraction pattern. (b) Net light intensity of the diffraction pattern, that is, the background coming from the Gaussian beam is subtracted.

Figure 9. Net intensity profile of the diffraction pattern presented in Figure 8b (a) in the horizontal (x-axis) (b) and vertical (y-axis) directions.

Figure 6 presents the location of the (first order) suppressions of the interference fringes which vary as a function of obstacle’s size. Herein, this dependence is shown for various positions (z2) of obstacles in a water drop. The analysis of these graphs shows that despite the strong relationship between the location of the suppressions of interference fringes and the size of the obstacle, determination of the obstacle size is burdened by a big uncertainty if the location (z2) of an obstacle in the water drop in the direction along the optical axis of Gaussian beam is not known. Fortunately, Figure 7a shows that there is a clear monotonic relation between the position of the obstacle inside the drop (z2) and the angular position of the diffraction maxima. This monotonic relationship is true for all diffraction orders; however, the figure shows it for the first two orders. It is obvious that such a monotonic relationship of the angular position of the diffraction

Figure 4 leads to two main conclusions. With an increase in size of the obstacle, angular positions of suppressed interference fringes are shifted toward their center. In contrast, the angular positions of the interference maxima and minima are almost stationary. On the other hand, Figure 5 shows the profile of the net intensity of interference fringes due to the disk-shaped obstacle with fixed radius. The subsequent insets (Figure 5a−d) depict this profile for different positions (z2) within the water drop. These positions are equally distributed along the optical axis of the Gaussian beam (z2). In these graphs, one can notice that the maxima and minima of the interference fringes reduce their “spatial frequency” if the obstacle moves in the direction of light propagation. Unfortunately, there is also clearly visible a shift of the angular positions of the suppressions of interference fringes as a function of the obstacle’s position along a light beam (z2). 3142



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maxima allows us to determine the position of the observed obstacles (z2) inside the drop (along beam of light). With this location known, the appropriate curve can be selected from the bunch of lines in Figure 6. Now, based on the angular position of the suppression of the interference fringes, the size of the examined obstacles can be determined. The above-described procedure of finding the obstacle’s size is made possible by the fact that the angular position of the maxima of interference fringes to a small degree depends on the obstacle’s size. This fact is shown in Figure 7b for two orders of interference maxima. The disc-shaped obstacle discussed so far can be a model of particles in the amorphous phase. It is clear that the crystals are better simulated by other shapes. As an example of such shape, a rectangle is chosen. Fortunately, the discussed model of experimental setup (Figure 1) also makes it possible to evaluate the characteristic dimensions of obstacle in this form. Figure 8 shows the diffraction pattern of 2 μm × 6 μm rectangle. The rectangle is located in the middle of a water drop and its longer edge is aligned horizontally. In order to better emphasize the details of the diffraction pattern, Figure 8b shows the same image as in Figure 8a, however, without the background coming from the Gaussian beam. In Figure 8b it can be seen that the diffraction maxima and minima are in the form of concentric circles. This is related to the symmetry of the phase distribution of outgoing Gaussian beam from a water drop. On the other hand, the angular position of the interference fringes suppressions of the first order (as well as higher orders) in the horizontal direction is considerably closer to the optical axis of the Gaussian beam than it is in the case of the vertical direction. Even better, it is seen in Figure 9, which presents the net intensity profile of the diffraction pattern presented in Figure 9b in the horizontal (x-axis) and vertical (y-axis) directions. The analysis above proves that the analysis of the diffraction pattern produced by the experimental setup shown in Figure 1 also enables estimation of shapes of obstacles. In actual cases, the particles can have other shapes than those shown in Figure 2a, but in each case, the diffraction image includes information about the shape of the particles; thus characteristic sizes of the particles can be determined in the same way as described above, that is, on the basis of the parameters of diffraction fringes in the direction of its longer and shorter size.

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ASSOCIATED CONTENT

S Supporting Information *

Theoretical background behind numerical calculation of the diffraction pattern due to the obstacle in the water drop illuminated by the Gaussian light beam, as schematically shown in Figure 1. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/ cg501696y.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported by Polish Ministry of Science and Higher Education, Grant No. I-3/501/17-3-1-774. REFERENCES

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4. CONCLUSIONS Results of the calculations presented above show that the analysis of diffraction pattern of the early stages of small object growth in the experimental system proposed in this paper gives the opportunity to evaluate both the position and the characteristic sizes regardless of whether the objects are amorphous particles or crystals. Admittedly, evaluation of these parameters using the proposed experimental setup is more burdensome and may be less accurate compared to the situation when a microscope is used; however, the proposed experimental setup offers additional benefits. First, the study of crystalline or amorphous particles growth on the order of single microns is carried in a relatively large volume. These conditions are difficult to achieve using a typical microscope. Second, it is possible to simultaneously observe crystals at different depths. This last feature of the presented system also enables continuous and simultaneous observation of a few crystals that move about in different directions and measuring parameters of their movements. 3143


Evaluation of Size and Shape of Crystalline and Amorphous Particles

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