Development of measuring diffusion coefficients by digital holographic interferometry in transparent liquid mixtures

M.G. He; S. Zhang; Y. Zhang; S. G. Peng

doi:10.1364/OE.23.010884

1. Introduction

The study of diffusivity is important in many fields such as chemical engineering, pollution control, biology and separation of isotopes. Knowledge of diffusivity is necessary for the design of chemical equipment and for the mass transfer studies [1,2]. Diffusion coefficients can be obtained by experimental methods or empirical correlations. However, experimental methods can give more accurate results than empirical correlations. Diffusion coefficients in transparent liquid mixtures can be measured by optical methods including conventional interferometry, holographic interferometry and electronic speckle pattern interferometry. Holographic interferometry is one of the most widely used techniques for diffusivity [3,4] because of requiring simpler elements and less effort than conventional interferometry [5] and being a whole-field technique [1,6,7]. Compared with holographic interferometry, the digital one is more popular because it doesn’t need wet processing, has no limits on taken frames and has the fast and simple nature.

The measurement of diffusion coefficients by digital holographic interferometry can be classified depending on mathematical models. There are four methods which are widely researched. The first one is using the distance of the peaks in concentration difference profile between two times to determine diffusion coefficient, which was put forward by Bochner and Pipman in double-exposure holographic interferometry [5]. The second one is using the location of the fringes in interference fringes, which was put forward by Gabelmann-Gray and Fenichel [8,9], and developed by Ruiz-Bevia et al [10]. The third one is using relationship between the deflection angle of object beam and the concentration distribution in diffusion cell, which was put forward by Chhaniwal, Anand, et al [11,12]. The last one is by fitting equations for the experimental concentration profiles at different times, which was put forward by Mialdun et al [13].

The second method used by Gabelmann-Gray and Fenichel has two disadvantages. One is that it requires the central bright fringe in the middle of the interference fringes [9], which is not always true because of a small wedge angle in the diffusion direction between the two windows in diffusion cell [5], or something else. The other is that it requires the time interval between two exposures to be short [10], in which situation it’s hard to detect the change of concentration accurately. Then Ruiz-Bevia et al had solved the problems by using integral formula and pairs of fringes of the same interference order [10]. But the method by Ruiz-Bevia et al is complex because it needs modifying the interface position between two solutions and calculating diffusion coefficients over and over. In the third method there’s no the reference beam [11,12], which makes the experimental setup simpler and possible to be an instrument. However, there is one backward. It’s hard to handle because it requires the deflection large enough to be detected accurately and small enough to keep the object beam collimated. The forth method can combine the measurement of diffusion coefficients and that of Soret coefficients. However, it’s more complex. The first method is relatively simpler to handle.

The method using the distance of the peaks in concentration difference (phase difference) profile, can be classified as several kinds depending on the experimental setup, such as multiple beam interferometric method [14], Michelson interferometric method [14], fringe projection method [14], lensless Fourier transform method [1,15], Mach-Zehnder interferometric method [16] and so on. As Chhaniwal et al [14] described, the former three methods all use the shift of the interference fringes between two times to determine the distance. Lensless Fourier transform method uses spherical wave as the reference beam, which makes it easier to get phase information of object beam by Fresnel-Kirchoff diffraction integral. In this method the phase difference between two times is obtained and a numerical reference wave-front tilted at a slight angle is used to convert the infinite fringes to finite ones, which has inflection points corresponding to the two peaks in concentration difference profile. Then the finite fringes are skeletalized for determination of the distance. If the one-dimensional phase difference profile can be obtained numerically, it can be more convenient and accurate for measurement of the distance. In the Mach-Zehnder interferometric method, a plane wave is used as the reference beam, which leads to that it’s convenient to pick up the phase information of object beam through filtering in frequency-domain, such as Mialdun et al [13] and He et al [16] did. In this method, it’s very easy for hologram analysis and to obtain the one-dimensional phase difference profile numerically.

Based on the above, the method of using the distance of two peaks in phase difference profile by digital holographic interferometry and Mach-Zehnder interferometric experimental setup is easy to handle, and the obtained one-dimensional phase difference profile makes it convenient and accurate to measure the distance. In this method there are two quantities to be measured for determination of diffusion coefficients, the diffusion time and distance between the peaks. In order to improve the accuracy of measurement by this method, it’s necessary to have appropriate procedures of hologram analysis and accurate determination of the diffusion time.

In the method there are two things to think about in the procedures of hologram analysis. One is the determination of the phase difference between two times, and the other is the unwrapping of phase difference. He et al [16] and Mialdun et al [13] have researched on these. They have different ideas on the procedures. The procedure for determination of phase difference used by He et al, is more convenient without a spectrum moving in Fourier domain to eliminate the carrier frequency f₀, compared with the one used by Mialdun et al. However, the unwrapping procedure used by Mialdun et al is simpler and has more accurate results than the one by He et al, which can be recognized through the unwrapping results in their papers. In a word, a combination of the procedures used by He et al and those used by Mialdun et al is made in this work.

In order to determine diffusion time accurately, the initial time for diffusion needs to be researched. The shift of the initial time in the method used by Bochner and Pipman was thought to be about 10 seconds [5]. In the method they chose freezing the solutions before putting them to contact, which cannot be used widely. In the experiment by Mialdun et al [17], the shift of the initial time was large to about 6 min in some condition and it was varying with concentration interval between solutions for diffusion. These mean that the shift of the initial time will be different depending on diffusion cell and the concentration interval between solutions. For example, a shift of the initial time with one minute will make a difference of 2.5% for diffusion coefficients, two minutes 5%, when the two exposure times are chosen at t₁ = 30min, t₂ = 50min. When the diffusion time is larger, the influence of the initial time is smaller. However, there is limitation for t₂ (t₂ > t₁) in order to satisfy the condition of one-dimensional infinite diffusion in the finite diffusion cell. At the same time, the time interval between two exposure times should be larger in order to detect the change of concentration accurately when diffusion time is larger, because the change of concentration between certain time interval will be smaller as time is increasing. Therefore, the first exposure time t₁ can’t be chosen very large, which maybe makes obvious influence on measurement results with the shift of initial time. In order to eliminate this uncertain error, the initial time needs to be determined. In the work by Mialdun et al [13], the initial time is determined by fitting equation for many experimental concentration profiles at different times through varying diffusion coefficient and the initial time. However, this fitting way is not suitable for the method used in this paper. And the problem hasn’t been discussed in detail as a separate topic, even in the work by Mialdun et al [13,17]. In this paper, the reason for the shift of initial time is explained, and two functions for minimization are given to determine the initial time.

With the two kinds of development for the method, the diffusion coefficient of KCl in water at the concentration of 0.33mol/L and 25 °C is measured.

2. Theory

In a one-dimensional infinite medium the solution of the diffusion equation at time t for two liquids initially separated at y = 0, with concentrations c₁ and c₂ is

c (y, t) = \frac{c_{1} + c_{2}}{2} + \frac{c_{1} - c_{2}}{\sqrt{π}} \int_{0}^{\frac{y}{2 \sqrt{D t}}} \exp (- η^{2}) d η .

where D is the diffusion constant with the difference of concentration c₁-c₂ being small. Then the concentration difference between two times t₁ and t₂ can be expressed as:

Δ c (y, t_{1}, t_{2}) = \frac{c_{1} - c_{2}}{\sqrt{π}} (\int_{0}^{\frac{y}{2 \sqrt{D t_{2}}}} \exp (- η^{2}) d η - \int_{0}^{\frac{y}{2 \sqrt{D t_{1}}}} \exp (- η^{2}) d η) .

With a plane object wave going through the diffusing solutions perpendicular to the diffusion direction, the information of concentration difference will be introduced into the phase of the object beam. Because refractive index of solutions is varying linearly with the concentration for small interval of concentration [5], the phase difference of object beam can be expressed as

Δ φ (y, t_{1}, t_{2}) = φ_{2} - φ_{1} = 2 π L Δ n / λ + Δ φ_{0} (t_{2} - t_{1}) = 2 π L m Δ c (y, t_{1}, t_{2}) / λ + Δ φ_{0} (t_{2} - t_{1}) .

where φ is the phase of object beam, L is the thickness of diffusing liquid, Δn is the difference of refractive index, λ is the wavelength of light in vacuum, m is the mean value of the derivative for the applied concentration range, and Δφ₀ is the phase difference without change of concentration. Because Δφ₀ is the same when y is changed, the phase difference has the same profile with the concentration difference. The maximum and minimum of the concentration difference (phase difference) between two times t₁and t₂ in the diffusion direction can be obtained by the formula

w = \sqrt{\frac{8 D \ln (t_{2} / t_{1})}{(1 / t_{1}) - (1 / t_{2})}} .

where w is the distance between the maximum and minimum of the concentration difference (two peaks in the concentration difference profile). Therefore, we get the formula to calculate diffusion coefficient

D = w^{2} \frac{t_{1} / t_{2} - 1}{8 t_{1} \ln (t_{1} / t_{2})} .

With the help of digital holographic interferometry, we can get the phase difference profile of object beam to obtain the distance w. Finally, we can calculate diffusion coefficient with the three parameters w, t₁ and t₂.

With one plane wave split into a reference beam and an object beam, we make the object beam go through the diffusion cell and interfere with the reference beam at some angle. Then we use a CCD camera with lens to record the hologram. Through analysis of the hologram, we can obtain the object beam.

The reference beam at the CCD plane can be expressed in complex amplitude, with z-direction perpendicular to the CCD plane

\begin{array}{l} R (x, y, t) = R_{0} (x, y, t) \exp [j ({\vec{k}}_{R} \vec{r} - ϕ_{R} (x, y, t))] \\ = R_{0} \exp [j (\frac{2 π \cos θ_{R x}}{λ} x + \frac{2 π \cos θ_{R y}}{λ} y - ϕ_{R})] . \end{array}

where R₀ is the real amplitude, k_R is the wave vector, ϕ_R is the projection at the CCD plane from the phase distribution in the plane perpendicular to wave vector k_R, θ_Rx is the angle between wave vector and x-direction, and θ_Ry is the angle between wave vector and y-direction. The object beam can be expressed as

\begin{array}{l} O (x, y, t) = O_{0} (x, y, t) \exp [j ({\vec{k}}_{O} \vec{r} - ϕ_{O} (x, y, t))] \\ = O_{0} \exp [j (\frac{2 π \cos θ_{O x}}{λ} x + \frac{2 π \cos θ_{O y}}{λ} y - ϕ_{O})] . \end{array}

The interference of the reference beam and the object beam will result in a spatially varying intensity distribution at the CCD plane. This hologram can be expressed as

\begin{array}{l} I (x, y, t) = O_{0}^{2} + R_{0}^{2} + O_{0} R_{0} \exp [j (2 π \frac{\cos θ_{R y}}{λ} y + 2 π \frac{\cos θ_{R x}}{λ} x + ϕ_{O} - ϕ_{R})] \\ + O_{0} R_{0} \exp [- j (2 π \frac{\cos θ_{R y}}{λ} y + 2 π \frac{\cos θ_{R x}}{λ} x + ϕ_{O} - ϕ_{R})] . \end{array}

Here, we choose y-direction parallel with the diffusion direction, making both θ_Ox and θ_Oy equal to 90 degrees. The hologram we get at time t, can be converted into the Fourier domain

\begin{array}{l} F {I (x, y)} = F {O_{0}^{2} + R_{0}^{2}} \\ + F {O_{0} R_{0} \exp [j (ϕ_{O} - ϕ_{R})]} \otimes F {\exp (j 2 π \frac{\cos θ_{R y}}{λ} y + j 2 π \frac{\cos θ_{R x}}{λ} x)} \\ + F {O_{0} R_{0} \exp [- j (ϕ_{O} - ϕ_{R})]} \otimes F {\exp (- j 2 π \frac{\cos θ_{R y}}{λ} y - j 2 π \frac{\cos θ_{R x}}{λ} x)} \\ = G_{0} (f_{x}, f_{y}) + G (f_{x} - \frac{\cos θ_{R x}}{λ}, f_{y} - \frac{\cos θ_{R y}}{λ}) + G^{*} (f_{x} + \frac{\cos θ_{R x}}{λ}, f_{y} + \frac{\cos θ_{R y}}{λ}) . \end{array}

where F means the Fourier transform, f_x and f_y are the frequency, and ⊗ means convolution. Because the reference beam and the object beam are plane waves, O₀ and R₀ don’t change with coordinate. G₀(f_x, f_y) is nearly located at origin of coordinate. If there is not concentration difference in the diffusion cell, ϕ_O-ϕ_R almost doesn’t change with coordinate and G(f_x-cosθ_Rx/λ, f_y-cosθ_Ry/λ) presents a point at (cosθ_Rx/λ, cosθ_Ry/λ) in Fourier domain. Thinking about the concentration difference in the diffusion cell, ϕ_O-ϕ_R is the function of y. So G(f_x-cosθ_Rx/λ, f_y-cosθ_Ry/λ) is actually a line in Fourier domain, and G*(f_x + cosθ_Rx/λ, f_y + cosθ_Ry/λ) is also a line. With appropriate angle θ_Rx and θ_Ry, the three parts G₀, G and G* are separated in Fourier domain. In a word, we can convert the hologram of object and reference beam into Fourier domain, obtain G or G* by appropriate filter, and get O₀R₀exp[j(2πycosθ_Ry/λ + 2πxcosθ_Rx/λ + ϕ_O-ϕ_R)] or O₀R₀exp[-j(2πycosθ_Ry/λ + 2πxcosθ_Rx/λ + ϕ_O-ϕ_R)] through inverse Fourier transform.

Because both θ_Ox and θ_Oy are equal to 90 degrees, which means the object beam is perpendicular to the CCD plane, the phase distribution of object beam φ is equal to ϕ_O. Then we can get the phase difference of object beam between two times t₁ and t₂ using the method by He et al [16].

\begin{array}{l} Δ φ = ϕ_{O}^{t_{2}} - ϕ_{O}^{t_{1}} - (ϕ_{R}^{t_{2}} - ϕ_{R}^{t_{1}}) \\ = Im [\ln (\frac{O_{0}^{t_{2}} R_{0}^{t_{2}} \exp [j (2 π \frac{\cos θ_{R y}}{λ} y + 2 π \frac{\cos θ_{R x}}{λ} x + ϕ_{O}^{t_{2}} - ϕ_{R}^{t_{2}})]}{O_{0}^{t_{1}} R_{0}^{t_{1}} \exp [j (2 π \frac{\cos θ_{R y}}{λ} y + 2 π \frac{\cos θ_{R x}}{λ} x + ϕ_{O}^{t_{1}} - ϕ_{R}^{t_{1}})]})] . \end{array}

This method is convenient without moving the spectrum from (cosθ_Rx/λ, cosθ_Ry/λ) or (-cosθ_Rx/λ, -cosθ_Ry/λ) to origin in Fourier domain.

The phase difference Δφ calculated by above function is wrapped, which means that it belongs to the range (-π, π). We need convert it into the continuous phase difference, which makes it convenient and accurate for determination of distance between the peaks in phase difference numerically. The continuous phase difference obtained by He et al [16], has larger deviation with the theoretical phase difference profile, which can be recognized in the paper. However, the unwrapping procedure by Mialdun et al [13], is simple and has good results. Therefore, we unwrap the phase difference to construct the continuous natural phase as Mialdun et al did. If the difference is less than π, the phase remains unchanged. If the phase of the latter pixel is more than π bigger than the former in y-direction, the phases of the latter pixel and all pixels behind are minus 2π. If the phase of the latter pixel is more than π smaller than the former in y-direction, the phases of the latter pixel and all pixels behind are plus 2π.

3. The initial time

Figure 1 shows the sketch of the diffusion cell used in this work. Firstly, the lighter solution at some concentration is filled into the diffusion cell and all capillary tubes for draining air. Then the heavier solution is filled into the tubes through liquid inlet with exhaust port (exhaust port and liquid outlet) closed and liquid outlet open until the condition shown as Fig. 1(a). Subsequently, the heavier solution is admitted slowly from below lifting the lighter solution through liquid inlet with exhaust port open and liquid out closed, which is shown in Fig. 1(b) under the ideal condition. However, there will be mixing process with bulk-flow when the heavier solution is filled into the diffusion cell through tubes, because the section of the diffusion cell is much larger than that of tubes. This mixing process will make the interface between two solutions into a mixing area, which is shown in Fig. 1(c).

Fig. 1 Sketch of the diffusion cell. (a) Before lifting the lighter solution. (b) Ideal condition in the lifting process. (c) Actual condition in the lifting process.

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Although the time of mixing between two solutions is short, the speed of mass transfer by mixing with bulk-flow is much faster than that by diffusion. When the shift of initial time due to mixing with bulk-flow is transferred to that due to diffusion, it would be large. Therefore, the shift of initial time needs to be considered in the measurement. If it counts to be zero (time recording starts) when the interface between two solutions is established, the initial time t₀ would be negative (t = t^count-t₀ with t^count the recorded time and t the actual diffusion time). In the diffusion cell used by this work, there is some lighter solution in tubes when it’s started to lift the lighter solution, which is shown in Fig. 1(a). Therefore, it takes time until the interface is established from starting to lift the lighter solution. If it counts to be zero when the heavier solution is admitted to lift the lighter solution, the initial time t₀ would be negative or positive depending on design of the diffusion cell and tubes.

Considering the influence of the initial time t₀, diffusion time t can be expressed as t = t^count – t₀, with t^count being the recorded time. Then the Eq. (4) can be expressed as

w = \sqrt{\frac{8 D (t_{1}^{count} - t_{0}) (t_{2}^{count} - t_{0}) \ln [(t_{2}^{count} - t_{0}) / (t_{1}^{count} - t_{0})]}{t_{2}^{count} - t_{1}^{count}}} .

In order to divide the initial time and diffusion coefficient, the Eq. (11) can be transferred into

w = \sqrt{8 D / Δ t} \sqrt{(t_{2}^{count} - t_{0}) (t_{2}^{count} - t_{0} - Δ t) \ln [(t_{2}^{count} - t_{0}) / (t_{2}^{count} - t_{0} - Δ t)]} .

where ∆t is equal to t₂^count-t₁^count. As ∆t is fixed, the distance w is varying with time t₂^count and there are two parameters D and t₀. In the curve of function w varying with time t₂^count, the parameter D determines the slope of the curve and the parameter t₀ determines the shift of the curve, which can be recognized through Eq. (12) and Fig. 2. On the condition of ∆t = 50 min and D = 1.845 × 10⁻⁹ m²/s, there are several curves with different initial time t₀, which is shown in Fig. 2(a). On the condition of ∆t = 50 min and t₀ = 0, there are several curves with different D, which is shown in Fig. 2(b).

Fig. 2 Influence of the initial time and diffusion coefficient on the curve of function w. (a) Influence of the initial time on the curve of function w. (b) Influence of diffusion coefficient on the curve of function w.

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Based on the above, diffusion coefficient D and the initial time t₀ are two separated parameters. Therefore, it can be used for determination of t₀ to minimize the deviation between the experimental curve of w and the calculated one by varying D and t₀, when ∆t is fixed. The function for minimization is expressed as

F_{1} = \sum_{i} {(w_{i}^{cal} - w_{i}^{\exp})}^{2} .

where w^cal is calculated by Eq. (12) and w^exp is measured in the experimental phase difference profile. When the initial time is determined in this way, the diffusion coefficient is obtained at the same time.

Considering the influence of the initial time t₀, the Eq. (5) can be expressed as

D = w^{2} \frac{(t_{1}^{count} - t_{0}) / (t_{2}^{count} - t_{0}) - 1}{8 (t_{1}^{count} - t_{0}) \ln [(t_{1}^{count} - t_{0}) / (t_{2}^{count} - t_{0})]} .

Equation (14) can be transferred into

D = \frac{w^{2} Δ t}{8 (t_{2}^{count} - t_{0}) (t_{2}^{count} - t_{0} - Δ t) \ln [(t_{2}^{count} - t_{0}) / (t_{2}^{count} - t_{0} - Δ t)]} .

Through Eq. (15), it can be found that the distance w is varying with t₂^count to keep the diffusion coefficient constant when ∆t is fixed. Supposing that the initial time t₀ is equal to 5 min and diffusion coefficient is equal to 1.845 × 10⁻⁹ m²/s, a series of distance w at different t₂^count can be obtained by Eq. (12) with ∆t being 50 min. Then the diffusion coefficients can be calculated with the series of distance w. If shift of the initial time isn’t taken into consideration, the calculation results of diffusion coefficient by Eq. (5) will be different as t₂^count is increasing, which is shown in Fig. 3. The same is for the situation t₀ = −5 min, which is also displayed in Fig. 3. If the initial time is taken into consideration, the results of diffusion coefficient will be constant as Fig. 3 shows.

Fig. 3 Contrast of calculation results with considering the initial time and without considering

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In Fig. 3, it can be found that calculation results of diffusion coefficient is varying with time without considering shift of initial time, when the initial time is not zero. Taking shift of initial time into consideration, the calculation results will be constant with the right initial time. Therefore, it can be used for determination of t₀ to minimize difference among calculation results of diffusion coefficient by varying t₀. The function for minimization is expressed as

F_{2} = \sum_{i} {(D_{i}^{cal} - {\bar{D}}^{cal})}^{2} .

where D^cal is the calculation result by Eq. (15) and

{\bar{D}}^{cal}

is the average value of different calculation results. Through minimization of the function expressed as (16), the average diffusion coefficient and initial time can be obtained at the same time.

4. Experiment

Figure 4 shows the schematic of the digital holographic interferometry setup employed for our measurement of diffusion coefficient. A spatially filtered and expanded He-Ne laser beam is split into a reference and an object beam. After passing through the diffusion cell perpendicular to the diffusion direction, the object beam interferes with the reference beam and makes a hologram, which is recorded by a CCD camera with lens. The CCD camera (FA-21-1M120), has 1024 × 1024 pixels with the pixel size of 7.4 × 7.4 μm². One pixel for the CCD camera corresponds to 43.79 um in the diffusion cell.

Fig. 4 Experimental setup for the measurement of diffusion coefficients

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The experimental cell is shown like Fig. 5. The diffusion cell has three disks with a rectangular cell to hold the diffusing liquids in the middle disk. The rectangular cell measures 1.2 cm × 1.2 cm × 6 cm with glass plates on two sides to allow the laser light to pass through. The diffusion cell is put in water bath to be kept at the constant temperature.

Fig. 5 Diffusion cell system: 1 exhaust port and liquid outlet, 2 Pt100 (Fluke), 3 liquid inlet, 4 liquid outlet, 5 outlet for water bath, 6 water bath, 7 diffusion cell, 8 inlet for water bath.

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There are some capillary tubes cycling in water bath for preheating liquid and these tubes are connected to the liquid inlet and have about half volume of the rectangular cell for diffusion. At first, the lighter solution at some concentration is filled into diffusion cell and all capillary tubes. Then the heavier solution with another concentration replaces the lighter liquid in the tubes through liquid inlet 3 and liquid outlet 4. After 20 minutes, the heavier solution is admitted slowly from below lifting the lighter liquid until the interface reaching the middle of the diffusion cell through liquid outlet 1 and liquid inlet 3. It counts to be zero when heavier solution starts to lift the lighter.

Water bath is provided and controlled by F33-EHJULABO. The fluctuation of temperature in the water bath is within 0.02 K. The temperature measured by the Pt100 (Fluke) has an uncertainty of ± 0.02 K. Considering both the fluctuation of temperature and the uncertainty of the temperature sensor Pt100, the total uncertainty for the temperature in the diffusion cell is within 0.04 K.

With this experimental setup, we measure the diffusion coefficient of KCl in water at the concentration of 0.33mol/L and 25 °C. The two kinds of concentration are 0.28 and 0.38 mol/L, as Ruiz-Bevia et al [10] chose.

5. Procedures of hologram analysis

The hologram is recorded as a picture every 5 minutes, as Fig. 6(a) shows. Each hologram is converted into Fourier domain, which is shown in Fig. 6(b). In the Fourier domain there are three parts, which include an ellipse and two lines. The ellipse corresponds to G₀ and noise, whereas the two lines correspond to G and G*. A rectangular filter is used to obtain G or G*. The size and location of the rectangular filter window is chosen manually to cover one line, as Fig. 6(c) displays. After filtering, an inverse Fourier transform is used to get the phase information of the object beam, O₀R₀exp[j(2πycosθ_Ry/λ + 2πxcosθ_Rx/λ + ϕ_O-ϕ_R)] or O₀R₀exp[-j(2πycosθ_Ry/λ + 2πxcosθ_Rx/λ + ϕ_O-ϕ_R)].

Fig. 6 Picking up of the phase information of object beam. (a) Hologram recorded by CCD. (b) Fourier transform of the hologram. (c) Picking up of the phase information of object beam.

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After getting the phase information of the object beam at times t₁ and t₂, the phase difference (interference phase) can be obtained with Eq. (10), which is shown in Fig. 7. Through the procedure expressed above, we can unwrap the phase difference into continuous phase like Fig. 8.

Fig. 7 Wrapped phase difference of object beam between two times

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Fig. 8 Unwrapped phase difference of object beam between two times

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After getting the continuous phase difference, we can use polynomial to fit the phase difference in y-direction (diffusion direction). The result for one profile in y-direction is shown in Fig. 9, in which we can find that the fitting curve and the original phase difference coincide very well. The fitting value in each pixel is obtained for determination of the peaks in the phase difference profile.

Fig. 9 Fitting of phase difference profile by polynomial in y-direction with red curve for the original and green one for the fitting

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After getting the location of the peaks at each x coordinate, we can get the distribution of the peaks of concentration difference (phase difference), which is shown in Fig. 10(a). Through this figure we can make the conclusion that the assumption of one-dimensional diffusion is satisfied in the experiment. Besides, we can get the distance w of the peaks in concentration difference profile at each x coordinate like Fig. 10(b). With the average in x-direction, we can get the average w for calculating the diffusion coefficient.

Fig. 10 (a) Distribution of the peaks of concentration difference in diffusion cell. (b) Distance between the two peaks.

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Because the size and location of the rectangular filter window is chosen manually depending on the figure in Fourier domain, the measurement result of the distance w is different between two times of analysis. In order to overcome the error, the analysis of each pair of holograms is treated twice to get average value of distance w. The average value is w, whereas the stand deviation is σ and the relative deviation is δw = σ/w. When the relative deviation is large, it means that noise has obvious influence on the hologram analysis. This phenomenon is always happened when the time interval between two exposure times is relatively short. When the relative deviation is small, it means that the hologram analysis has good accuracy.

6. Experimental results

We measure the diffusion coefficient of KCl in water at 0.33mol/L and 25 °C. The hologram is recorded as a picture every 5 minutes. Through any two pictures, we can get a distance w between the two peaks of phase difference. Each pair of holograms has been treated twice to get the average. The average value is w, whereas the stand deviation is σ and the relative deviation is δw = σ/w. The measurement results are shown in Table 1. The left is for the condition ∆t = 40 min, whereas the right is for the condition ∆t = 50 min. The relative deviation in Table 1 is less than 0.5%, which means that the results have good accuracy.

Table 1. Distance of two peaks of phase difference at different time intervals.

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With a series of distance w at different t₂^count, the diffusion coefficient can be determined by minimizing function F₁ when ∆t is fixed. This way to determine diffusion coefficient is called the first way. And the way by minimizing function F₂ is called the second way. With some pairs of w and t₂^count at fixed ∆t, it can lead to a pair of diffusion coefficient and initial time with the first way. The same is for the second way. In order to certify the stability of results, numbers of pairs of w and t₂^count are changed from 20 to 9, as Table 2 and Table 3 show. The measurement results of diffusion coefficient by both first and second way are shown in Table 2 and Table 3. The distribution of diffusion coefficients calculated by two ways is shown in Fig. 11.

Table 2. Diffusion coefficients calculated at ∆t = 40 min.

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Table 3. Diffusion coefficients calculated at ∆t = 50 min.

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Fig. 11 Diffusion coefficients calculated by two ways when ∆t = 40 and ∆t = 50min

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With the average of all results for diffusion coefficients, we obtain the final diffusion coefficient of KCl in water at the concentration of 0.33mol/L and 25 °C, which is 1.844 × 10⁻⁹ m²/s with the uncertainty of ± 0.012 × 10⁻⁹ m²/s. The relative uncertainty of diffusion coefficient is less than 1%. The contrast of our measurement result with literature data is shown in Table 4. The deviation of our result with Harned and Gosting is positive and less than 0.3%, whereas the deviation with Szydlowska and Ruiz-Bevia, who used holographic interferometry, is negative and about 1%.

Table 4. Contrast of our result and literature data for the diffusion coefficient of KCl in water at 0.33mol/L and 25 °C.

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7. Conclusion

As for the measurement of diffusion coefficient in transparent liquids by digital holographic interferometry, there are many kinds depending on mathematical model and experimental setup. The method of using the distance of the peaks in concentration difference profile to determine diffusion coefficient by Mach-Zehnder interferometric experimental setup, is easy to handle. In order to improve the accuracy and convenience of hologram analysis in this method, a combination of procedures for hologram analysis used by Mialdun et al and those used by He et al, has been made. After getting continuous phase difference of object beam between two times, the experimental phase difference profile is fitted by polynomial. It turns out that the fitting curve has good coincidence with the experimental. This makes it convenient and accurate to determine the distance of the peaks in concentration difference profile.

In order to determine diffusion time accurately, the shift of the initial time needs to be considered. When the heavier solution is injected into the diffusion cell to lift the lighter solution, there will be mixing process with bulk-flow because the section of the diffusion cell is much larger than the tubes for filling. Although the time of mixing between two solutions is short, shift of the initial time will be large when the influence of mixing with bulk-flow on the initial time is transferred to that of diffusion. In the method used by this work, two functions for minimization are given to determine the initial time.

With the development, diffusion coefficient of KCl in water at 0.33mol/L and 25 °C has been measured. The measurement result of diffusion coefficient is 1.844 × 10⁻⁹ m²/s with the uncertainty of ± 0.012 × 10⁻⁹ m²/s. Our measurement has more similar result with other methods than holographic interferometry. The relative uncertainty of diffusion coefficient in our experiment is less than 1%. The total uncertainty of temperature in our experiment is within ± 0.04 K.

Acknowledgments

This work was supported by the Major State Basic Research Development Program of China (973 Program No. 2015CB251502).

References and links

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∆t = 40 min					∆t = 50 min
t₁^count	t₂^count	w	σ	δw	t₁^count	t₂^count	w	σ	δw
min	min	pixels	pixels	%	min	min	pixels	pixels	%
35	75	121.306	0.006	0.00	35	85	125.222	0.085	0.07
40	80	131.851	0.049	0.04	40	90	135.825	0.015	0.01
45	85	141.487	0.061	0.04	45	95	145.821	0.026	0.02
50	90	149.924	0.056	0.04	50	100	154.150	0.060	0.04
55	95	157.028	0.058	0.04	55	105	161.736	0.011	0.01
60	100	165.187	0.049	0.03	60	110	169.946	0.137	0.08
65	105	172.518	0.201	0.12	65	115	177.749	0.122	0.07
70	110	180.916	0.016	0.01	70	120	184.644	0.026	0.01
75	115	187.808	0.013	0.01	75	125	190.853	0.035	0.02
80	120	193.535	0.281	0.15	80	130	196.545	0.024	0.01
85	125	199.943	0.431	0.22	85	135	205.314	0.119	0.06
90	130	203.728	0.050	0.02	90	140	208.498	0.091	0.04
95	135	212.093	0.178	0.08	95	145	215.057	0.029	0.01
100	140	215.948	0.047	0.02	100	150	219.482	0.084	0.04
105	145	223.010	0.088	0.04	105	155	226.016	0.084	0.04
110	150	226.236	0.096	0.04	110	160	230.245	0.180	0.08
115	155	229.988	0.536	0.23	115	165	235.539	0.009	0.00
120	160	236.052	0.335	0.14	120	170	242.575	0.089	0.04
125	165	242.710	0.047	0.02	125	175	245.097	0.049	0.02
130	170	251.459	0.435	0.17	130	180	252.137	0.055	0.02
135	175	250.834	0.542	0.22
140	180	260.263	0.077	0.03

	Numbers of pairs of w and t₂^count
	20	19	18	17	16	15	14	13	12	11	10	9
First way
D	0.1856	0.1840	0.1837	0.1840	0.1850	0.1855	0.1849	0.1852	0.1843	0.1855	0.1850	0.1844
× 10⁻⁸ m²/s	0.1856	0.1840	0.1837	0.1840	0.1850	0.1855	0.1849	0.1852	0.1843	0.1855	0.1850	0.1844
t₀	16.38	15.98	15.89	15.97	16.21	16.33	16.19	16.27	16.07	16.33	16.21	16.11
min	16.38	15.98	15.89	15.97	16.21	16.33	16.19	16.27	16.07	16.33	16.21	16.11
Second way
D	0.1849	0.1839	0.1837	0.1839	0.1846	0.1848	0.1844	0.1846	0.1839	0.1846	0.1841	0.1835
× 10⁻⁸ m²/s	0.1849	0.1839	0.1837	0.1839	0.1846	0.1848	0.1844	0.1846	0.1839	0.1846	0.1841	0.1835
t₀	16.17	15.95	15.90	15.95	16.08	16.14	16.05	16.08	15.94	16.09	15.99	15.88
min	16.17	15.95	15.90	15.95	16.08	16.14	16.05	16.08	15.94	16.09	15.99	15.88

	Numbers of pairs of w and t₂^count
	20	19	18	17	16	15	14	13	12	11	10	9
First way
D	0.1849	0.1844	0.1847	0.1841	0.1842	0.1845	0.1843	0.1848	0.1847	0.1854	0.1835	0.1840
× 10⁻⁸ m²/s	0.1849	0.1844	0.1847	0.1841	0.1842	0.1845	0.1843	0.1848	0.1847	0.1854	0.1835	0.1840
t₀	16.15	16.02	16.09	15.92	15.96	16.03	15.98	16.10	16.08	16.23	15.82	15.92
min	16.15	16.02	16.09	15.92	15.96	16.03	15.98	16.10	16.08	16.23	15.82	15.92
Second way
D	0.1846	0.1843	0.1844	0.1840	0.1841	0.1843	0.1841	0.1845	0.1843	0.1847	0.1833	0.1837
× 10⁻⁸ m²/s	0.1846	0.1843	0.1844	0.1840	0.1841	0.1843	0.1841	0.1845	0.1843	0.1847	0.1833	0.1837
t₀	16.04	15.97	16.00	15.90	15.92	15.96	15.93	16.00	15.98	16.06	15.77	15.84
min	16.04	15.97	16.00	15.90	15.92	15.96	15.93	16.00	15.98	16.06	15.77	15.84

Result in our work D ( × 10⁻⁹ m²/s)	The same concentration with our experiment
1.844 ± 0.012	D ( × 10⁻⁹ m²/s)	Deviation with literature data	Author (time)	Method
	1.842	0.11%	Harned et al (1949) [18]	Conductance measurement
	1.839	0.27%	Harned et al (1949) [18]	Onsager-Fuoss Theory
	1.841	0.16%	Gosting (1950) [19]	Gouy interference
	1.864 ± 0.037	−1.07%	Szydlowska et al (1982) [20]	Holographic interferometry
	1.86 ± 0.02	−0.86%	Ruiz-Bevia et al (1985) [10]	Holographic interferometry
	Different concentration with our experiment
	D ( × 10⁻⁹ m²/s)		Author (time)	Method
	1.837 (0.274mol/L)		Rard et al (1980) [21]	Rayleigh interferometry
	1.830 (0.2mol/L), 1.867 (0.5mol/L)		Lobo et al (1998) [22]	Conductance measurement

∆t = 40 min					∆t = 50 min
t₁^count	t₂^count	w	σ	δw	t₁^count	t₂^count	w	σ	δw
min	min	pixels	pixels	%	min	min	pixels	pixels	%
35	75	121.306	0.006	0.00	35	85	125.222	0.085	0.07
40	80	131.851	0.049	0.04	40	90	135.825	0.015	0.01
45	85	141.487	0.061	0.04	45	95	145.821	0.026	0.02
50	90	149.924	0.056	0.04	50	100	154.150	0.060	0.04
55	95	157.028	0.058	0.04	55	105	161.736	0.011	0.01
60	100	165.187	0.049	0.03	60	110	169.946	0.137	0.08
65	105	172.518	0.201	0.12	65	115	177.749	0.122	0.07
70	110	180.916	0.016	0.01	70	120	184.644	0.026	0.01
75	115	187.808	0.013	0.01	75	125	190.853	0.035	0.02
80	120	193.535	0.281	0.15	80	130	196.545	0.024	0.01
85	125	199.943	0.431	0.22	85	135	205.314	0.119	0.06
90	130	203.728	0.050	0.02	90	140	208.498	0.091	0.04
95	135	212.093	0.178	0.08	95	145	215.057	0.029	0.01
100	140	215.948	0.047	0.02	100	150	219.482	0.084	0.04
105	145	223.010	0.088	0.04	105	155	226.016	0.084	0.04
110	150	226.236	0.096	0.04	110	160	230.245	0.180	0.08
115	155	229.988	0.536	0.23	115	165	235.539	0.009	0.00
120	160	236.052	0.335	0.14	120	170	242.575	0.089	0.04
125	165	242.710	0.047	0.02	125	175	245.097	0.049	0.02
130	170	251.459	0.435	0.17	130	180	252.137	0.055	0.02
135	175	250.834	0.542	0.22
140	180	260.263	0.077	0.03

Development of measuring diffusion coefficients by digital holographic interferometry in transparent liquid mixtures

Abstract

1. Introduction

2. Theory

3. The initial time

4. Experiment

5. Procedures of hologram analysis

6. Experimental results

7. Conclusion

Acknowledgments

References and links

Cited By

Figures (11)

Tables (4)

Equations (16)

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