CONDENSED MATTER DIFFUSION COEFFICIENT OF POTASSIUM DIHYDROGEN PHOSPHATE USING HOLOGRAPHIC INTERFEROMETRY* MONA MIHAILESCU1, RALUCA AUGUSTA GABOR2 1 “Politehnica” University from Bucharest, 2 ICECHIM Bucharest, E-mail: mona_m@physics.pub.ro Received November 4, 2009 The holographic interferometry technique associated with the fringe processing in the Fourier plane permits the real-time monitoring due to digital recording with great resolution and is used to detect the optical path length variations in a transparent media and consequently the temporal and spatial distribution of the refractive index in liquid solutions with concentration gradient. A new method based on the auto-correlation coefficients, the convolution and shift theorems is introduced to visualize the displacement of the fringe pattern in regions with different concentration in isotropic and anisotropic conditions. The experimental results have been used in the diffusion process modeling and to calculate the diffusion coefficient of the potassium dihydrogen phosphate in water. Key words: holographic interferometry, potassium dihydrogen phosphate, pollution, diffusion coefficient. 1. INTRODUCTION Potassium Dihydrogen Phosphate (PDP) is a highly water-soluble salt which is often used as a fertilizer on the field, followed by the changes of the nutrient balance in soil and consequently in the surface water. For this reason, its diffusion coefficient in water is important for research and in many applications: chemical engineering, biology, pollution control. It isn't a toxic substance, but its secondary effect is the increase of the plants growth rate in the surface waters, process known with the name of eutrophication [1]. The decomposition of the plants depletes the supply of oxygen which leads to the death of fish and other sub-aquatic animals. Eutrophication leads to daily variations in the oxygen concentration and the water PH [2]. Other places where the diffusion coefficient of fertilizers is important, are the constructed wetlands. The information about nutrients distribution gives a better prediction of the treatment efficiency and leads to a better optimization of the size and the geometry of the wetlands [3]. * Paper presented at the “Optoelectronic Techniques for Environmental Monitoring” (OTEM2009), September 30–October 2, 2009, Bucharest, Romania. Rom. Journ. Phys., Vol. 56, Nos. 3–4, P. 399–410, Bucharest, 2011 400 Mona Mihailescu, Raluca Augusta Gabor 2 The holographic interferometry technique is an advanced optical tool used to investigate liquid media. From the fringes movement, processed in the Fourier plane, we deduce the temporal behavior of the PDP in water, using the shift theorem. The solution of Fick's law in that given situation is employed to make a link between the displacement in the holographic images, the temporal evolution of the concentration gradient and the diffusion process. A new procedure, based on the auto-correlation and convolution theorem is proposed here to calculate the displacement from one image to another and, respectively, to distinguish the zones with significant refractive index variations. Auto-correlation coefficients measure the correlation between observations at different times [4]. In our case, the observations are fringe patterns recorded with a specific time interval between them and through the calculus of the auto-correlation coefficients we compare the concentration values [5]. 2. EXPERIMENTAL PROCEDURES Potassium dihydrogen phosphate (KH2PO4) is a synthesized active ingredient included in the pesticide class of fungicide type. The end-use product is a crystalline powder containing 100% active ingredient. It is easily dissolved by the rain and arrives in the surface waters from field. In lakes, the nutrient level gradually increases over time. In the lectic systems, the oxygen and the light, the factors proper for photosynthesis, decrease with the depth and the rate of the algae growth is also reduced under the euphotic zone. But, if some nutrients get in there, the plants grow over the normal rate and the oxygen depletion and the eutrophication appear also at great depths. To know the depth where phosphate has a harmful concentration, we must know its diffusion coefficient. Our samples were prepared from a known quantity of pure distilled water in a rectangular cuvette (0.5x0.5x4cm) on which we added a given quantity from PDP powder on its surface, to achieve the desired concentration. In these conditions the maximum concentration value can be 20%, which is obtained without agitator (at 25°C). The experimental setup for the holographic interferometry measurements is based on the configuration of Mach-Zehnder interferometer with this sample in one arm. The object laser beam ( λ = 632.8nm ) traverses this sample at the middle of the cuvette and is superposed with the reference beam. On the CCD sensor (7µm pitch, 30 frames/s) appears a fringe pattern. We start the recording process in the moment when the powder is added, till no movement is present, which corresponds with stabilized known concentration (its refractive index is measured using Abbe refractometer). The refractive index of any solution depends on its concentration. This relation is different from one solution to another and was determined experimentally 3 Diffusion coefficient of potassium dihydrogen phosphate 401 for our substances using given concentrations (prepared separately) followed by the fitting process (see Fig. 1a and 1b). We chose the fitting curve of the polynomial type of two or three order. The residuals obtained besides experimental data show a standard deviation of about 0.001 in the correspondence between the refractive index and the concentration values (see Fig. 1c and 1d). So, starting with any value of the refractive index, we can estimate the value of the concentration with an acceptable accuracy and consequently the distribution in time and space of the concentration values. This link between the refractive index and the concentration is an improving method to calculate this dependency besides the previous method [6], where the dependency was supposed linear. a) c) b) d) Fig. 1 – Refractive index vs. concentration starting from experimental points and fitted with polynomial of a) two and b) three order; In c) and d) are the histograms of the residuals obtained after fitting. 402 Mona Mihailescu, Raluca Augusta Gabor 4 3. SIMULATIONS The simulation process starts with the experimental fringes image processing. From the movie obtained on the CCD, we collect hundred of images, frame to frame, which are firstly indexed and the gray level, the brightness and the contrast are adjusted with the same scale. From these images, imported in MATLAB, we constitute the data base for the refractive index determination in two different situations: from one image in distinct regions and from consecutive images in the same region, in order to extract the spatial and temporal evolution of the refractive index. The unwrapped phase in the Fourier plane contains information about the displacements in the object plane which is recovered using the shift theorem. According to the light-wave theory, the definition of the light path length and the holographic interference principle, when a beam passes through the sample, the variation of the refractive index in any point in the mass transfer region can be obtained using a reference image, from the equation [7]: ∆n( x, y ) = n( x, y ) − n( x0 , y0 ) = ∆ ( x, y ) ⋅ λ d ⋅i (1) where n( x0 , y 0 ) is the refractive index of the water, ∆( x, y ) is the fringe shift map d is the thickness of the sample, i is the interfringe. In the natural systems, the refractive index variation is done due to the diffusion process, when the concentration of different substances in water, changes its values in time and space. Starting with the experimental results, we create in MATLAB a model for the diffusion process based on Fick's second law: ∂c( x, t ) ∂ 2 c ( x, t ) = Kd ∂t ∂x 2 (2) where K d is the diffusion coefficient (m2/s) and c is the concentration (mol/m3). To solve this parabolic problem defined on the bounded domains, we use the Finite Element Method. The boundary conditions are of the Dirichlet type on the bottom and upper sides and of the Newman type on the left and right sides. Here we consider only the isotropic process. The colorbars from the right sides of Fig. 2a and 2b are coded in concentration values between 0% - bottom and 50% - upper. Analytically, to solve the eq. (2), we use Green's theorem [8] and Green's function for the classical diffusion equation, which is Gaussian spreading in time x2 + y 2 1 Gd ( x, y, t ) = exp − . The PDP concentration in a fixed position 4 Dτ 4πD τ at t + τ is given by the convolution between the concentration at t and the Green function [9]: c( x, t + τ) = c( x, t ) ⊗ Gd ( x, t ) (3) 5 Diffusion coefficient of potassium dihydrogen phosphate a) 403 b) Fig. 2 – The cuvette with vertical concentration gradient after 10s when the diffusion coefficient is a) K d = 9 ⋅ 10 −6 m 2 / s and b) K d = 9 ⋅ 10 −3 m 2 / s . We assume that the diffusion coefficient is constant in time and space, and the interval τ is the time between two consecutive frames recorded in the experimental setup. The fringe changes can be obtained by real-time monitoring and there is no time delay. The reference image is recorded with simple distilled water in cuvette. 4. IMAGE PROCESSING USING AUTO-CORRELATION COEFFICIENTS, CONVOLUTION AND SHIFT THEOREMS The Fourier analysis plays a key role in achieving the direct numerical extraction data from the experimental recorded images about their displacement. A translation in the fringe pattern due to refractive index changes, is perceived in the complex function obtained after Fourier transform. In this way, we use the shift theorem with the function t0 ( x, y ) associated with the fringe pattern recorded with simple distilled water in the cuvette and the functions t j ( x, y ) associated with the fringe pattern recorded at the frame j after the beginning of the diffusion process. We denote the Fourier transform T0 ( p, q ) = F (t0 ( x, y )) and T j ( p, q ) = F (t j ( x, y )) (4) A simple displacement in the fringe pattern from frame to frame introduced by the optical path variations leads us to the phase function changes in the Fourier plane [10]: F (t j +1 ( x, y )) = F (t j ( x + a , y + b )) = F (t j ( x, y )) ⋅ exp −i2π ( pa + qb ) (5) where p, q are the spatial frequency in the Fourier plane. The shift theorem (equation 5) was computed to determine the vector that translates the fringe patterns, with the components ( a, b) . Mona Mihailescu, Raluca Augusta Gabor 404 6 The inverse Fourier transform of the power spectrum (PSD) of the recorded intensity: PSD( p, q) = F [t ( x, y ) ] 2 (6) gives the complex auto-correlation function, as implied by the Wiener-Khintchine theorem [11], which returns the correlation coefficients for the pairs of data values: r ( x, y ) = { F −1 F [ t ( x , y ) ] 2 } − t ( x, y ) t ( x, y ) 2 − t ( x, y ) 2 (7) 2 written in the normalized form. The auto-correlation method is commonly used in the image processing field to identify the magnitude of the displacement. The convolution theorem between two functions associated with two images recorded in different conditions, F (t j ⊗ tl ) = T j ⋅ Tl (8) is applied to distinguish the relation between the shift from the mean intensity at a time and the shift from the mean intensity at some time later. 5. RESULTS AND DISCUSSIONS a) c) b) d) Fig. 3 – Experimental fringes obtained when the sample is: a) the simple cuvette; b) the cuvette with water. Cuvette with water and PDP at frame number c) 13 and d) 14 from the beginning of the diffusion process. The numbers from the axis are the pixels number. 7 Diffusion coefficient of potassium dihydrogen phosphate 405 In Fig. 3a is shown the fringe pattern recorded with the simple cuvette in the holographic setup and in Fig. 3b is shown the fringe pattern when we added simple distilled water. The two dimensional optical path length distribution introduced by the sample is obtained by the fringe shift in comparison with the fringe pattern without PDP and the refractive index is calculated using equation 1. These optical path lengths are converted in the refractive index distribution and then in concentration values, using the calibration curve. The fringe patterns from Fig. 3c and 3d are recorded at two consecutive frames with the refractive index gradient present inside the cuvette. One can observe the curvature of the fringes in Fig.3c and 3d, with greater radius in Fig. 3c. Fig. 4 – Two profile lines in the same region in the interference pattern from Fig. 3c -o- and from Fig. 3d -x-. The link between the refractive index gradient and the curvature of the light path was done elsewhere based on the solution of the Fick's law in a given geometrical condition [12]. The displacements between fringes, plotted in Fig. 4, indicate a refractive index variation of ∆n = 0.326 ⋅ 10 −3 between two consecutive frames in the same region. In Fig. 5 are shown the amplitude functions images in the Fourier plane starting with the fringe pattern from Fig. 3. Additional information can be extracted using the Fourier analysis; we will present three complementary analysis. The curvature of the fringes from Fig. 3c and d is distinguished in the Fourier plane through the appearance of the circular pattern around the central peak, with a better visibility and increased value of radius in Fig. 5c. The radius of the circular pattern in the Fourier plane is linked with the radius in the image plane through the Airy diffraction pattern. Our calibration process reveals that the radiuses in the fringe pattern are in the range from a few meters to a few hundred of centimeters with a minimum in the first part of the experiment, when the concentration gradient is maxim. Mona Mihailescu, Raluca Augusta Gabor 406 a) 8 b) c) Fig. 5. Amplitude images in the Fourier plane starting with the fringe pattern from a) Fig. 3b, b) Fig. 3c, c) Fig. 3d. A displacement in a given position of the fringe pattern, can be seen in the Fourier plane using the phase profile accordingly with the shift theorem (equation 5). It is well known that after Fourier transform, the phase function is wrapped, with values between −π and π . In Fig. 6 is plotted a line from the unwrapped phase function in the Fourier plane starting with the images from Fig. 3. In the same region, the displacement of the fringe is calculated for three different moments with the reference in the case of simple cuvette in the experimental setup. From these dates we calculate the displacement vector with the components (a,b) having the signification from the equation 5. As one can see, the position for the first order peaks are the same in Fig. 5a, Fig. 5b and Fig. 5c due to the same interfringe in all patterns from Fig. 3. Nevertheless, some characteristic features are distinguished in the Fourier plane through the differences between images. In Fig. 7a is the difference between Fig. 5b and 5a, in Fig. 7b is the difference between Fig. 5c and 5a. The black and white regions represent the maximum differences in negative (black) or positive (white) 9 Diffusion coefficient of potassium dihydrogen phosphate 407 values, while the gray regions represent the regions with the same values in both figures. All matrices are normalized before the calculation of the differences. Fig. 6 – Phase shift for c - simple cuvette, w - cuvette with distilled water, f13 - the sample with DPD in water at 13-th frame, f14 - the sample with DPD in water at 14-th frame. a) b) Fig. 7 – The differences in the Fourier plane between a) Fig. 5b and 5a, c) Fig. 5c and 5a. Fig. 8 – Variation of the fringe pattern auto-correlation coefficient in the cases: w - simple distilled water in the cuvette, f13 - the sample with PDP in water at 13-th frame, f14 - the sample with PDP in water at 14-th frame. Mona Mihailescu, Raluca Augusta Gabor 408 10 Using the auto-correlation function given by the equation 7, we calculate the 2D auto-correlation map starting with the fringe pattern from Fig. 3b, c and d. In Fig. 8 are plotted the auto-correlation profiles from the middle of these maps. The central peaks from the zero order auto-correlation coefficients are removed. The fringe patterns were 1024x1024 pixel size. The r value depends by the number of the pixels under consideration and their location [13]. For the inhomogeneous conditions, we introduced a new method to visualize the regions where the concentration (and consequently the refractive index) has different values from point to point and from frame to frame. We apply the convolution theorem (equation 8) using like reference the fringe pattern recorded with simple distilled water in the experimental setup. a) b) Fig. 9 – The convolution maps between the reference fringe pattern with simple distilled water in the cuvette and the fringe pattern with concentration gradient inside the cuvette from frame a) 13-th and b) 14-th. The convolution between two fringe patterns is performed to contour the zone with strong concentration gradient. In Figs. 9a and b are shown the convolution map between the reference fringe pattern and the fringe pattern from 13-th and 14-th frame, respectively. The cropped regions from the right side of Fig. 9 are studied to measure the magnitude of the path length variations and the obtained values are till 1.4π. This maximum value corresponds with the maximum refractive index variations of 0.442 ⋅ 10−3 , at the same frame in two neighboring pixels. These changes were due to refractive index variation between the reference (for distilled water) and 1.347 which corresponds to the concentration values between 0% and 21.72%. For the diffusion coefficient, a mean value of K d = 1.27 ⋅ 10−9 m 2 / s calculated in pdetoolbox from MATLAB, matches the experimental values obtained for concentration at two different moments in the same region and at the same moment in two different regions. 11 Diffusion coefficient of potassium dihydrogen phosphate 409 4. CONCLUSIONS The diffusion coefficient of the potassium dihydrogen phosphate in water was investigated using a combination of techniques, experimental and also numerical: real-time holographic interferometry, refractometry, fringe analysis in Fourier plane, the shift theorem, the convolution and auto-correlation operations, the diffusion process modeling. Measurements using solutions with given concentrations in Abbe refractometer give us the experimentally dependence between concentration and the refractive index. The accuracy of about 0.001 is used to pass from refractive index values to concentration values. The fringe processing in the Fourier plane offers an alternative method to reveal complementary information about the curvature of the fringes and about the differences in the optical path introduced by the sample, using the amplitude maps in Fourier plane and the shift theorem. For the image processing, a method based on the auto-correlation coefficients and the convolution theorem is introduced here to distinguish the displacement and to measure it in the regions with strong concentration gradient, with one pixel resolution. The experimental data base is used to model the processes and to calculate the diffusion coefficient. 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