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
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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.
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Mona Mihailescu, Raluca Augusta Gabor
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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.
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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.
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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.
We verified that the holographic interferometry method completed with the
digital processing of the fringe pattern is a cheap and easy, non-destructive and
non-intrusive laser-based technique for measurements of the temporal and spatial
refractive-index distribution in a transparent solution with a concentration gradient.
In the future we must study the same process in a deeper and thicker cuvette.
Acknowledgements. The experiments described in this paper were carried out using
equipments acquired by Contract 4/CP/I/11.09.2007 PNCDI II "Capacities" and the research has been
(partially) supported by the ANCS-UEFISCSU grant ID 1556 / 2009.
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