New method to measure liquid diffusivity by analyzing an instantaneous diffusion image

Licun Sun; Weidong Meng; Xiaoyun Pu

doi:10.1364/OE.23.023155

1. Introduction

Diffusion is a process by which matter is transported from one part of a system to another part due to molecules’ thermal energy under the influence of a concentration gradient [1, 2]. The diffusion coefficient (D) is a basic physical property data in chemical engineering and knowledge about it is needed for mass transfer studies and in many other fields such as crystal growth, biological systems, pollution control and separation of isotopes and so on [3–7]. Diffusion phenomena can appear in all gaseous, liquid and solid materials. To measure the D values of liquids is more difficult than the values of gases and solids [8, 9]. Many physical and chemical methods are employed for finding the liquid diffusivity experimentally, for instance, holographic interferometry [10], Taylor dispersion method [11] and the fluorescent molecule tracing [12], each with their characteristic advantages and disadvantages. The several common disadvantages in these methods include (1) take long time (>2h) to obtain the experimental data, (2) cannot observe the diffusion process directly, (3) need very stringent optical conditions.

Some emerging methods can solve these problems partly. For example, the experimental equipment of Moiré deflectometry method [13] is simple and it is low sensitive to external disturbances. To overcome these disadvantages, we have designed and fabricated an asymmetric liquid-core cylindrical lens (ALCL), which is able to measure refractive index (RI) of filled liquids in the way of spatial resolution along the axis of ALCL. ALCL is used as both the diffusion pool and the core imaging device, in which a dynamic gradient distribution of RI forms along the diffusion direction because of molecule diffusion. Once a thin liquid layer with a fixed RI (n_c) is selected to be imaged in measurements, a shifting sharp focal point related to the layer will appear on CCD. Based on Fick’s second law [1], only one instantaneous image in the diffusion process is required to figure out the binary liquid D value, because the gradient distribution of RI, n(Z), is implied by the width of the diffusion image. It takes only 20 micro-seconds to capture one instantaneous image, less than 10 seconds to get information about n(Z), and work out D values by using an analytic program designed by ourselves. Compared with the traditional methods [10–13] of which several hours are needed for measuring a single D value, the method is characterized by faster measurement, direct observation of diffusive process, and easy operation.

Ethylene glycol (EG), featured by non-corrosive and non-crystalline, has the effect of absorbing carbon dioxide and dehumidification, researches find that EG is of important application in the field of environmental protection in recent years [14]. Accurate D values of EG in water at five different temperatures from 288.15 K to 308.15 K were measured by analyzing instantaneous diffusion images in this paper, and the results were consistent well with the values measured by using holographic interferometry [15] and Taylor dispersion method [16].

2. Principle and method

2.1 Imaging principle

ALCL is composed by sticking two different cylindrical lenses together in order to reduce spherical aberration to the minimum. Acted as both a diffusion pool and a key imaging device in the D value measurement, the imaging principle for measuring D with ALCL is shown in Fig. 1. When collimated light beams passing though the ALCL, if the ALCL is filled with one uniform liquid, the focal lengths at different heights (Z_i) of ALCL are the same, a CCD chip positioned at the focal plane will receive a sharp line image, as shown in Fig. 1(a). If two kinds of liquids with different RIs are loaded into the ALCL in successively, the refractive capabilities of the upper and lower parts of ALCL are different. When collimated light beams in upper part is imaged clearly on the CCD, the lower part of the beams is a diffused image, as shown in Fig. 1(b). Once the two liquids are contacted each other, the diffusion process commences. A dynamic gradient distribution of concentration for the two solutions is formed gradually along the Z-axis, corresponding to a measureable RI gradient distribution. If the CCD is fixed at a specific position, where the collimated light beams will be imaged clearly for a thin liquid layer with an explicit RI value (say n = n_c), an image with “beam waist” pattern shall appear on the CCD chip, as shown in Fig. 1(c). The narrowest width of the image is determined mainly by diffraction limit, however, the width of other area in the image is a function of RI value of liquid layer, which will be described in next part of this paper. Since liquid diffusion is a dynamic process, the “beam waist” pattern will drift slowly along the Z-axis, which is recorded visually as shown in Visualization 1, when the chemical EG diffusing in pure water.

Fig. 1 Illustrations of the imaging principle for ALCL filled with different liquids. (a) Filled with uniform liquid of RI = n. (b) Filled with two different liquids, n₁<n₂. (c) A RI gradient distribution of the filled liquid is formed along Z-axis, n₁<n₂ = n_c<n₃<n₄, the dynamic process is showed in Visualization 1.

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2.2 Formula derivation for measuring RI

The top view of ALCL is showed as Fig. 2. The RI of liquid filled in the ALCL must be known before the measurement of D value. Let the focal length and RI of liquid filled in the ALCL be f_i and n_i, respectively, based on paraxial imaging Gaussian formula [17], the iterative relations between f_i and n_i can be written as,

f_{i} = \frac{R_{4} S_{4}}{n_{0} R_{4} + (n_{0} - 1) S_{4}} + d_{3} + d_{2} / 2,

S_{4} = \frac{n_{0} R_{3} S_{3}}{n_{i} R_{3} + (n_{i} - n_{0}) S_{3}} - d_{3},

S_{3} = \frac{n_{i} R_{2} S_{2}}{n_{0} R_{2} + (n_{i} - n_{0}) S_{2}} - d_{2},

S_{2} = \frac{n_{0} R_{1}}{n_{0} - 1} - d_{1} .

Where R₁ = 20.0 mm, R₂ = R₃ = 17.0 mm, and R₄ = 37.6 mm are the curvature radii of the four refraction surfaces, respectively; d₁ = d₂ = d₃ = 3 mm are the wall thickness and cavity thickness, respectively; material of cylindrical lens is K9 glass (n₀ = 1.5163 at λ = 589 nm). With the help of a digital translation stage, the RI = n_i of liquid filled in ALCL can be figured out by measuring the focal length f_i. The standard deviation in measuring RI is less than 0.0001 and in [18], Q. Li et al have described the measurement in detail.

Fig. 2 Top view of the designed ALCL. Yellow arrows indicate the beams focus exactly on the CCD plane after passing the ALCL filled with liquid of RI = n_c; Red arrows indicate the beams project on the CCD plane with a width of W_i after passing the ALCL filled with liquid of RI = n_i<n_c; Blue arrows are the same as the red arrows, but the ALCL filled with liquid of RI = n_i>n_c.

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Let h be the width of collimated light beams, f_c be the focal length of the ALCL filled with liquid of RI = n_c, f_i be the focal length of the ALCL filled with liquid of RI = n_i. If the CCD is fixed at the focal plane of the ALCL filled with liquid of RI = n_c, as shown in Fig. 2, the collimated beams passing the ALCL filled with liquid of RI = n_i>n_c (or n_i<n_c) will project a width W_i on the CCD plane. From the view of geometrical optics, the image width (W_i) and focal length (f_i) satisfies with

\frac{h / 2}{f_{i}} = \frac{W_{i} / 2}{| f_{i} - f_{c} |} .

Where h is a known value; f_c can be measured accurately by using the method mentioned before; W_i is the image width that can be read out straightforwardly by a software built in the CCD camera; thus, f_i can be calculated by Eq. (2). Inserting f_i to Eq. (1), n_i, the RI of liquid filled in ALCL can be figured out. Since W_i is varied with Z-axis as shown in Fig. 1(c), W_i = W_i(Z). After f_i = f_i(Z) is determined by measuring W_i = W_i(Z), n_i = n_i(Z), that is, the spatial distribution of RI is determined completely by only one diffusion image.

2.3 Calculation theory for diffusion coefficient

Assuming binary solutions involved in diffusion to be A and B, diffusion direction to be Z-axis, and C(Z, t) to be the mass fraction of A in B at diffusion time t and position Z. Let D be the diffusion coefficient, based on Fick’s second law [1], C (Z, t) satisfies with

\frac{d C (Z, t)}{d t} = D \frac{d^{2} C (Z, t)}{d Z^{2}} .

Assuming the initial concentrations to be C₁ and C₂ on each side of the contact interface (Z = 0) before the diffusion beginning. According to the boundary and initial conditions, the solution of Eq. (3) is the form of Gauss error function [19],

C (Z, t) = \frac{C_{1} + C_{2}}{2} + \frac{C_{1} - C_{2}}{2} e r f (\frac{Z}{2 \sqrt{D t}}) .

Where,

e r f (u) = \frac{2}{\sqrt{π}} \int_{0}^{u} e x p (- t^{2}) d t

. Let “erfinv” be the inverse error function, and suppose the function relation between solution concentration and its RI to be

C (Z, t) = g [n (Z, t)]

, Eq. (4) can be rewritten as

D = \frac{Z^{2}}{e r f i n v^{2} {[g [n (Z, t)] - \frac{C_{1} + C_{2}}{2}] / [\frac{C_{1} - C_{2}}{2}]} \cdot 4 t} .

The concrete form of

g [n (Z, t)]

is pre-determined by experimental method.

In actual measurements, due to the intermolecular attraction, the contact interface cannot be an exact horizontal plane which leads to a difficulty in positioning the contact interface exactly, i.e. the position of Z = 0. So the position Z, a value relative to the contact interface, will have a deviation ΔZ. Equation (5) can be revised to

Z = 2 \sqrt{D} e r f i n v {[g [n (Z, t)] - \frac{C_{1} + C_{2}}{2}] / [\frac{C_{1} - C_{2}}{2}]} \cdot \sqrt{t} - Δ Z .

In the method of instantaneous analysis, a diffusion image is recorded by a CCD chip at any time t. The spatial distribution of liquid concentration C(Z, t) caused by diffusion can be worked out by analyzing the spatial distribution of RI, n(Z, t), which is implied by the recorded image, as described in section 2.2. Let $2 \sqrt{D t}$ be a constant, taking linear regress analysis between Z_i and $e r f i n v {[g [n (Z, t)] - \frac{C_{1} + C_{2}}{2}] / [\frac{C_{1} - C_{2}}{2}]}$ , supposing the linear coefficient to be a, thus $a = 2 \sqrt{D t}$ . The diffusion coefficient to be measured can be expressed by,

D = (a)^{2} / 4 t,

which is independent on the deviation ΔZ.

3. Experimental setup

Figure 3 is the experimental setup. A 2-mW green laser beam (λ = 543.5 nm) was attenuated by a neutral density filter, and then the attenuated beam was expanded and collimated by a microscope objective lens ( × 10) and a double convex lens (f = 150 mm). An adjustable slit was used to control the width of the collimated beam. After passing though the ALCL filled with liquid, the light reached a CCD with 2448 × 2058 pixels and 3.45 × 3.45 µm² per pixel size. The CCD, connected to a PC via a USB, was driven by a translation stage with 0.001 mm minimum scale to adjust the imaging plane position slightly, which made the observation for a diffusion image convenient. The coordinates of arbitrary pixel on an image can be gotten and shown on PC screen by the software built in the CCD camera.

Fig. 3 Schematic of the measurement setup.

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The cross section of the ALCL was showed in Fig. 2 and its height was 50 mm. The bottom of the ALCL was sealed off with rubber membrane. The denser solution EG (chemical purity grade) was injected to ALCL’s lower half part using a digital syringe (RSP02-C, produced by Ruichuang Electronic Technology Co., Ltd) firstly, waiting for 5-10 minutes to damp liquid convection, and then the distilled water was introduced to the ALCL’s upper half part along the inner wall of ALCL by the same digital syringe with a slow speed (2.5ml/ min), making sure no obvious convection current in the two liquids. The injection of distilled water was taken about 2 minutes, the time of two solutions starting to contact was defined as the onset of diffusion (t = 0).

The CCD was set along Y-axis at a position between two focal planes where the ALCL was filled with water and EG, respectively. The CCD was also set along Z-axis at the position where the contacting interface of two solutions can be imaged on the bottom of the CCD. To reduce the interface turbulence caused by liquid injection, diffusion images were taken 20 minutes after the diffusion process beginning.

4. Experimental results and discussion

4.1 Preparatory works

4.1.1 Experimental relationship between the solution concentration and its RI

In order to calculate the diffusion coefficient D by Eq. (5) at different temperatures, the experimental relation between the concentration and RI of EG aqueous solution at different temperatures, $C (Z, t) = g [n (Z, t)]$ must be known in advance. To do it, all of experiments were carried out in an air-conditioned room. EG aqueous solutions were put to use when their temperatures were equal with ambient temperatures. Eight groups of EG aqueous solutions with different mass fractions were prepared using a precision electronic auto-balance with 0.0001g precision, and their RIs were measured by an Abbe Refractometer at different temperatures. The results were shown in Table 1, and the fitting relations between solution concentrations and related RIs were listed at the last row of Table 1, showing a good linear relationships whose correlation indexes were better than 0.9998.

Table 1. The experimental relationship between the concentration and the RI of EG aqueous solution at different temperatures

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4.1.2 Relationship between the image width and the RI filled in ALCL

Both the experimental and theoretical methods have been used to find the relationship between the image width and the RI of liquid filled in the ALCL. For the experimental method, distilled pure water of RI = 1.3334, measured by the Abbe Refractometer at 298.15 K, was injected into ALCL to calibrate the imaging system. A CCD was fixed at the position where the image appeared on the CCD was of the narrowest width, when the ALCL was filled with the liquid of RI = n_c = 1.3391. Groups of alcohol aqueous solutions were prepared and their RIs were measured by the Abbe Refractometer. When aqueous solutions with different RIs were filled in the ALCL in sequence, the images recorded by the CCD were shown in Fig. 4, which indicated that the image width decreased with the rise of RI, if RI is smaller than n_c = 1.3391, otherwise the image width increased with the rise of RI. The image widths (W_i) were measured and plotted in Fig. 5 by the dots.

Fig. 4 The experimental images for ALCL filled with uniform liquid with different RIs in the case that CCD was fixed at a position where the image appeared on the CCD was narrowest only when the ALCL was filled with the liquid of RI = n_c = 1.3391. (a) RI = 1.3334, (b) RI = 1.3349, (c) RI = 1.3369, (d) RI = 1.3391, (e) RI = 1.3429, (f) RI = 1.3489, (g) RI = 1.3607.

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Fig. 5 The relation between W_i and n_i. The dots are experimental data, and the line is calculation value.

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For the theoretical method, the RI of liquid in the Eq. (1) is substituted by n_i = n_c = 1.3391, the calculated focal length is f_c = 52.475mm. Setting the entrance slit be h = 7mm (experimental condition), according to Eq. (1) and Eq. (2), the image width (W_i) varied with liquid RI (n_i) is figured out, the result is shown in Fig. 5 by the solid line, which is in good consistent with experimental data. The experimental data is fitted linearly by the Least Square method, the fitting result is expressed by the conditional function as

n = {\begin{cases} - 0.0273 W + 1.3395,^{} n < n_{c}, \\ ^{^{}} 0.0276 W + 1.3388,^{} n > n_{c} . \end{cases}

4.2 Measurement of diffusion

In the whole experiment of EG diffusing in pure water, the CCD was fixed at the position where the width of image was the narrowest when ALCL was filled with liquid of RI = n_c = 1.3391. Both ambient and solution temperatures were 298.15 K in the first experiment. The whole diffusion process was shown in Visualization 1, while a typical diffusion image at the time of t = 1800s was shown in Fig. 6.

Fig. 6 One of the instantaneous images at the time of t = 1800s when chemical EG diffusing in pure water. The thick arrow indicates the position where the width of image was the narrowest referred to the liquid layer of RI = n_c = 1.3391.

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Data pairs of (Z_i, W_i)s were read out automatically by a software built in the CCD camera. The data pairs (Z_i, W_i)s were converted into (Z_i, n_i) based on Eq. (8), and then (Z_i, C_i) based on the formula of C = 9.8478n −13.130 in the 4th column of Table 1, so the spatial distributions of RI and concentration were obtained, those were shown in Fig. 7.

Fig. 7 The spatial distribution of RI and concentration of EG diffusing in pure water at the time of t = 1800s. Z_i is the distance between the position researched and the interface, the solid line is for eye-guide only.

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Let C₁ = 0, C₂ = 1, t = 1800s, and $C (Z_{i}, t) = g [n (Z_{i}, t)] = C (Z_{i}) = g [n (Z_{i})]$ = 9.8478*n(Z_i) −13.130, Eq. (6) can be expressed as

Z_{i} = 84.85 \sqrt{D} e r f i n v [27.2611 - 19.6956 \times n (Z_{i})] - Δ Z .

Z_i, and

e r f i n v [27.2611 - 9.6956 \times n (Z_{i})]

are fitted linearly by the least square method, the linear coefficient a = 0.2882 is obtained, and then D = 1.1536 × 10⁻⁵cm²/s is figured out by Eq. (7), which is a litter bit smaller than the literature value [15], 1.189 × 10⁻⁵cm²/s, measured by the method of holographic interferometry. It is worth noting that the total time cost on measuring D value is about 7 second, including 20ms for taking an instantaneous image, and 6.297 second for data-processing that can be shorten further. Therefore, fast measurement is the significant feature of the presented method.

Besides the instantaneous image shown in Fig. 6, other eight diffusion images, recorded by an interval of 5 minutes, are shown in Fig. 8. Using the same techniques in processing of Fig. 6, we have calculated D values based on the eight diffusion images, and the calculated results are listed in Table 2.

Fig. 8 “Beam waist” patterns recorded by a CCD at different diffusion moments.

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Table 2. The experimental D values measured by analyzing instantaneous images at different diffusion times

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It is clear that the calculated D values are stable basically in the period of 1200 to 3600s, the mean and standard deviations of D values are 1.1378 and 0.0353 × 10⁻⁵cm²/s, respectively. A larger value at 1200s, D = 1.1932 × 10⁻⁵cm²/s may be caused by the interface turbulence, while a smaller value at 3600s, D = 1.0866 × 10⁻⁵cm²/s is probably caused by the reading error of the image width.

4.3 Verification

For the same diffusion images shown in the Fig. 8, the diffusion coefficient (D) can also be figured out by different method. The narrowest positions (Z_i) of instantaneous images, marked by arrows in Fig. 8, drift with time (t_i) due to dynamic diffusion process. Since the narrowest position corresponds to a liquid layer with a fixed RI, n_c = 1.3391 in the experiment, g [n_c (Z, t)] in Eq. (6) is a constant that is g[n_c (Z, t_i)] = 9.8478*n(Z_i) −13.130 = 0.057. Let C₁ = 0, C₂ = 1, Eq. (6) can be expressed as

Z_{i} = 2 \sqrt{D t_{i}} e r f i n v (0.8856) - Δ Z .

The experimental data of Z_i varied with t_i are listed in the Table 3, where N represents the number of pixels between the “waist” and the diffusion interface, which are read out by the software built in the used CCD, and value Z_i is obtained by multiplying N with a pixel size. Z_i and $\sqrt{t_{i}}$ are fitted linearly by the least square method, the fitting result can be expressed as

Z_{i} = 74.466 \sqrt{t_{i}} - 506.126 (μ m) .

The linear correlation coefficient is 0.998. Comparing Eq. (10) with Eq. (11), we obtain the D value of EG diffusing in water at temperature 298.15 K, that is D = 1.1087 × 10⁻⁵ cm²/s. The value is closed to the mean value calculated by analyzing an instantaneous diffusion image, the reliability of the instantaneous image analysis is thus verified.

Table 3. The data of position Z_i varied with diffusion time

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4.4 D values at different temperatures

Liquid diffusion process is very sensitive to temperature. For the EG diffusing in pure water, the dependence of D value on temperature has been measured by using the instantaneous image analysis, and the result is shown in Table 4. Similar to our expectation, the measured D value increases with temperature, and the D values obtained by us are almost the same as that of infinite dilution diffusion coefficient measured by using Taylor dispersion method [16] at temperatures of 298.15, 303.15 and 308.15K. The functional dependence of the infinite dilution D value on temperature is given by the Arrhenius equation [20, 21] expressed as

\ln (D) = \frac{- E}{R T} + A .

Where E is the diffusion activation energy, R is the gas constant and A is a constant value. Our measured D values are well consistent with the Arrhenius equation as showed in Fig. 9, the correlation index is 0.9939. The measured activation energy is 2.098 Kcal/mol.

Table 4. The measured D value of EG diffusing in water at different temperatures

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Fig. 9 Arrhenius equation fitted by experimental datum

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4.5 Error analysis and discussion

According to Eq. (7), the deviation in measuring D value can be analyzed by the error transfer formula expressed as

| Δ D | = \sqrt{{(Δ D)}_{t}^{2} + {(Δ D)}_{a}^{2}} = \sqrt{{(\frac{\partial D}{\partial t})}^{2} {(Δ t)}^{2} + {(\frac{\partial D}{\partial a})}^{2} {(Δ a)}^{2}} .

The deviation of

{| Δ D |}_{t} = a^{2} Δ t / 4 t^{2}

is caused by the error in measuring diffusion time Δt. The distilled water is injected to the ALCL slowly in the diffusion experiment to avoid the convection current, and the moment that the first drop of water meet with EG solution is defined as the onset of diffusion process. But the diffusion mechanism when the amount of water is rare is not the same as that when the water is filled with the upper part cell. Therefore, there is a deviation in determining the diffusion onset time which is assumed to be Δt = 30s. Let t = 1200s and a = 0.2393, the calculated deviation of D value is

{| Δ D |}_{t}

= 0.0298 × 10⁻⁵cm²/s, which is the largest

{| Δ D |}_{t}

for the D values in Table 2. The deviation of

{| Δ D |}_{a} = a Δ a / 2 t

is composed by two parts, the error in measuring RI of liquid

{| Δ D |}_{n a}

and the error in measuring the image width

{| Δ D |}_{w a}

,

{| Δ D |}_{a} = {| Δ D |}_{n a} + {| Δ D |}_{w a}

. The accuracy of Abbe Refractometer, which is used to measure the RIs of experimental solutions, is Δn = 0.0002. Based on the experimental formula listed on the 4th column of Table 1, the concentration deviation caused by the value of Δn = 0.0002 is ΔC = 9.8478Δn = 0.0019, thus Δa = 0.0061, leading to a calculated deviation of D value

{| Δ D |}_{n a}

= 0.0489 × 10⁻⁵ cm²/s. The widths (W_i) of images are read out automatically by a software built in the CCD camera with one pixel reading error, so the error (ΔW) of width is 3.45 µm which produces a RI deviation Δn = 0.00009 according to Eq. (8), leading to a calculated deviation of D value

{| Δ D |}_{w a}

= 0.0224 × 10⁻⁵cm²/s. Therefore, the deviation of D value caused by the value a is about

{| Δ D |}_{a}

= 0.0713 × 10⁻⁵ cm²/s.

Combining the above error analysis, the total deviation of D value is $| Δ D |$ = 0.0773 × 10⁻⁵ cm²/s, and the main deviation is from the error in measuring the concentration of diffusion liquid. A higher precision measuring tool for RIs is necessary and useful to reduce ${| Δ D |}_{n a}$ , and a selected diffusion image with a longer diffusion time may decrease ${| Δ D |}_{t}$ to the minimum.

This method is based on the determination of the gradient distribution of RI, and the standard deviation in measuring RI is about 0.0001, therefore, the method does not work for two fluids with similar refractive indices. The gradient distribution of RI decreases as the diffusion process goes on, which also reduces measurement accuracy, so the diffusion images after 60 minutes are not collected and analyzed. The sensitivity of this method at present is inferior to holographic interferometry method [22] and Taylor dispersion method [23], and is approximate with Moiré deflectometry method [13]. However, the characteristics of the method, such as quick measurement, direct observation of diffusive process and easy operation, are irreplaceable by other measurement methods.

5. Conclusion

We introduce a new method for measuring liquid diffusion coefficient rapidly in the paper. The key optical device of the method is an ALCL that is used as both the diffusion cell and imaging lens. Owing to the spatially resolved ability of the ALCL in measuring liquid RI, the distribution of the liquid concentration caused by diffusion process can be recorded by an instantaneous diffusion image, and then the D value is calculated on the base of Fick’s second law. The experimental time cost on measuring a D value is shortened greatly from several hours in traditional method to a few seconds in the new method. The experiments of EG diffusing in pure water have been carried out, which demonstrates that the new method is feasible, and it opens a door for measuring liquid diffusion coefficient rapidly.

Acknowledgments

This work was supported by the National Science Foundation of China (Grant NO. 11164033 and 61465014).

References and links

1. J. Crank, The Mathematics of Diffusion (Oxford University, 1975).

2. R. Taylor and R. Krishna, Multicomponent Mass Transfer (Wiley & Sons Inc., 1993).

3. E. L. Cussler, Diffusion: Mass Transfer in Fluid Systems, 3rd ed. (Cambridge University, 2009).

4. N. Weiss, T. G. van Leeuwen, and J. Kalkman, “Simultaneous and localized measurement of diffusion and flow using optical coherence tomography,” Opt. Express 23(3), 3448–3459 (2015). [CrossRef] [PubMed]

5. C. Angstmann and G. P. Morriss, “An approximate formula for the diffusion coefficient for the periodic Lorentz gas,” Phys. Lett. A 376(23), 1819–1822 (2012). [CrossRef]

6. S. Lang, T. J. Kazdal, F. Kühl, and M. J. Hampe, “Diffusion coefficients and VLE data of aqueous phosphoric acid,” J. Chem. Thermodyn. 68, 75–81 (2014). [CrossRef]

7. Y. Zou, Z. Shen, X. Chen, Z. Di, and X. Chen, “An integrated tunable interferometer controlled by liquid diffusion in polydimethylsiloxane,” Opt. Express 20(17), 18931–18936 (2012). [CrossRef] [PubMed]

8. X. Liu, S. K. Schnell, J. Simon, P. Krüger, D. Bedeaux, S. Kjelstrup, A. Bardow, and T. J. H. Vlugt, “Diffusion coefficients from molecular dynamics simulations in binary and ternary mixtures,” Int. J. Thermophys. 34(7), 1169–1196 (2013). [CrossRef]

9. X. Liu, A. Bardow, and T. J. H. Vlugt, “Maxwell-Stefan diffusivities and velocity cross-correlations in dilute ternary systems,” Principles Diffusion Theory 16, 81 (2011).

10. V. K. Chhaniwal, A. Anand, S. Girhe, D. Patil, N. Subrahmanyam, and C. S. Narayanamurthy, “New optical techniques for diffusion studies in transparent liquid solutions,” J. Opt. A, Pure Appl. Opt. 5(5), S329–S337 (2003). [CrossRef]

11. T. C. Chan and W. K. Tang, “Diffusion of aromatic compounds in nonaqueous solvents: a study of solute, solvent, and temperature dependences,” J. Chem. Phys. 138(22), 224503 (2013). [CrossRef] [PubMed]

12. C. T. Culbertson, S. C. Jacobson, and J. Michael Ramsey, “Diffusion coefficient measurements in microfluidic devices,” Talanta 56(2), 365–373 (2002). [CrossRef] [PubMed]

13. K. Jamshidi-Ghaleh, M. T. Tavassoly, and N. Mansour, “Diffusion coefficient measurements of transparent liquid solutions using Moiré deflectometry,” J. Phys. D Appl. Phys. 37(14), 1993–1997 (2004). [CrossRef]

14. K. Y. Suh and R. Langer, “Microstructures of poly (ethylene glycol) by molding and dewetting,” Appl. Phys. Lett. 83(8), 1668–1671 (2003). [CrossRef]

15. J. Fernández-Sempere, F. Ruiz-Beviá, J. Colom-Valiente, and F. Más-Pérez, “Determination of diffusion coefficients of glycols,” J. Chem. Eng. Data 41(1), 47–48 (1996). [CrossRef]

16. M. H. Wang, A. N. Soriano, A. R. Caparanga, and M. H. Li, “Mutual diffusion coefficients of aqueous solutions of some glycols,” Fluid Phase Equilib. 285(1-2), 44–49 (2009). [CrossRef]

17. M. Born and E. Wolf, Principles of Optics (Cambridge University, 2003).

18. Q. Li, X.-Y. Pu, R.-F. Yang, and Y. Zhai, “Measurement of diffusion coefficient of liquids by using an asymmetric liquid-core cylindrical lens: observing the diffusion process directly,” Chin. Phys. Lett. 31(5), 054203 (2014). [CrossRef]

19. W. Jost, Diffusion in Solids, Liquids, Gases (Academic, 1960).

20. D. F. Othmer and M. S. Thakar, “Correlating diffusion coefficients in liquid,” Ind. Eng. Chem. 45(3), 589–593 (1953). [CrossRef]

21. E. D. Snijder, M. J. M. te Riele, G. F. Versteeg, and W. P. M. van Swaaij, “Diffusion coefficients of several aqueous alkanolamine solutions,” J. Chem. Eng. Data 38(3), 475–480 (1993). [CrossRef]

22. A. Anand, V. K. Chhaniwal, and C. S. Narayanamurthy, “Diffusivity studies of transparent liquid solutions by use of digital holographic interferometry,” Appl. Opt. 45(5), 904–909 (2006). [CrossRef] [PubMed]

23. G. D’Errico, O. Ortona, F. Capuano, and V. Vitagliano, “Diffusion coefficients for the binary system glycerol + water at 25°C. a velocity correlation study,” J. Chem. Eng. Data 49(6), 1665–1670 (2004). [CrossRef]

T n *C / %*	288.15 K	293.15 K	298.15 K	303.15 K	308.15 K
0.0	1.3344	1.3340	1.3334	1.3331	1.3324
10.0	1.3441	1.3437	1.3434	1.3425	1.3420
20.0	1.3542	1.3538	1.3535	1.3525	1.3520
30.0	1.3646	1.3643	1.3638	1.3625	1.3623
40.0	1.3750	1.3746	1.3739	1.3725	1.3720
50.0	1.3859	1.3850	1.3846	1.3830	1.3826
60.0	1.3953	1.3949	1.3939	1.3925	1.3918
70.0	1.4055	1.4046	1.4044	1.4025	1.4016
Fitting expression	C = 9.7748n-13.040, R² = 0.9998.	C = 9.8337n-13.116, R² = 0.9999.	C = 9.8478n-13.130, R² = 0.9999.	C = 10.0312n-13.369, R² = 0.9999.	C = 10.0596n-13.402, R² = 0.9999.

t(s)	1200	1500	1800	2100	2400	2700	3000	3300	3600
a(cm)	0.2393	0.2597	0.2882	0.3029	0.3358	0.3497	0.3723	0.3858	0.3956
D( × 10⁵cm²/s)	1.1932	1.1243	1.1536	1.0926	1.1749	1.1324	1.1552	1.1273	1.0866

t/s	1200	1500	1800	2100	2400	2700	3000	3300	3600
$\sqrt{t_{i}}$	34.6410	38.7298	42.4264	45.8258	48.9898	51.9615	54.7723	57.4456	60.0000
N	894	984	1064	1136	1202	1265	1328	1389	1443
Z_i /μm	3084.30	3394.80	3670.80	3919.20	4146.90	4364.25	4581.60	4792.05	4978.35

T(K)	288.15	293.15	298.15	303.15	308.15
D( × 10⁵cm²/s)	0.8272	0.9417	1.1378	1.2831	1.4409
D^a ( × 10⁵cm²/s)	–	–	1.102	1.241	1.381

T n *C / %*	288.15 K	293.15 K	298.15 K	303.15 K	308.15 K
0.0	1.3344	1.3340	1.3334	1.3331	1.3324
10.0	1.3441	1.3437	1.3434	1.3425	1.3420
20.0	1.3542	1.3538	1.3535	1.3525	1.3520
30.0	1.3646	1.3643	1.3638	1.3625	1.3623
40.0	1.3750	1.3746	1.3739	1.3725	1.3720
50.0	1.3859	1.3850	1.3846	1.3830	1.3826
60.0	1.3953	1.3949	1.3939	1.3925	1.3918
70.0	1.4055	1.4046	1.4044	1.4025	1.4016
Fitting expression	C = 9.7748n-13.040, R² = 0.9998.	C = 9.8337n-13.116, R² = 0.9999.	C = 9.8478n-13.130, R² = 0.9999.	C = 10.0312n-13.369, R² = 0.9999.	C = 10.0596n-13.402, R² = 0.9999.

New method to measure liquid diffusivity by analyzing an instantaneous diffusion image

Abstract

1. Introduction

2. Principle and method

2.1 Imaging principle

2.2 Formula derivation for measuring RI

2.3 Calculation theory for diffusion coefficient

3. Experimental setup

4. Experimental results and discussion

4.1 Preparatory works

4.1.1 Experimental relationship between the solution concentration and its RI

4.1.2 Relationship between the image width and the RI filled in ALCL

4.2 Measurement of diffusion

4.3 Verification

4.4 D values at different temperatures

4.5 Error analysis and discussion

5. Conclusion

Acknowledgments

References and links

Supplementary Material (1)

Cited By

Figures (9)

Tables (4)

Equations (16)

Optics Express