Double liquid-core cylindrical lens utilized to measure liquid diffusion coefficient

Weidong Meng; Yan Xia; Fangxi Song; Xiaoyun Pu

doi:10.1364/OE.25.005626

1. Introduction

Diffusion is the net movement of molecules or atoms from a region of high concentration to a region of low concentration caused by the thermal energy [1–3]. The study of diffusion is important in chemical engineering, biological systems, pollution control, separation of isotopes and other fields [4–7]. There are many well established methods [8–14] for the determination of liquid diffusion coefficient (D), interferometry [8–11] and Taylor dispersion [13-14] are the most widely used techniques for diffusivity studies among them. Those methods need very stringent experimental conditions, expend long measurement times (5000–20000s). To overcome the disadvantages in these methods, a liquid-core cylindrical lens used as both diffusion pool and key imaging element has recently been designed and used to measure liquid D values [15]. The liquid-core cylindrical lens is able to measure the refractive index (RI) of liquid filled in the core area along the cylindrical lens’s axis, which is the precondition for measuring D values. To measure D values accurately, liquid-core cylindrical lens is required to have high RI sensitivity, short depth of field [16], and small spherical aberration (SA) [17]. RI sensitivity and depth of field determine the resolvable minimum RI, while SA is the main aberration influencing the image quality, reducing SA from a liquid-core cylindrical lens is the key work in the present paper.

We fabricate firstly a symmetric liquid-core cylindrical lens based on the previous experience in designing liquid-core cylindrical lens [15,18], which guarantees the resolvable minimum RI is better than 0.0002 for most liquids filled in the symmetric liquid-core cylindrical lens. However, similar to the lens designed previously [15,18,19], the symmetric liquid-core cylindrical lens can only eliminate SA at a fixed liquid RI, SA is large for other liquids with different RI values. This shortcoming may spoil the diffusion image since the liquid RI is different along the diffusion direction. To reduce SA of the symmetric liquid-core cylindrical lens filled in different liquid, an aplanatic double liquid-core cylindrical lens (DLCL) is invented and reported in this paper. The front lens of the DLCL is filled with the liquid, of which RI is required to be measured in the way of spatial resolution. The rear lens of the DLCL is used as an aplanatic component, either the position or degree of the SA caused by the front lens can be regulated by selecting the liquid, of which RI is known and filled in the rear liquid core. Equipped with the aplanatic DLCL, two methods have been applied to measure the D value of 0.33mol/L KCL aqueous solution diffusing at temperature 298.15K, the first method derives D value from calculating the drift rate of diffusion image, while the second method obtains D value by analyzing an instantaneous diffusion image. The measured values are in good consistent with the literature value [20,21] that is a conventional criterion of 0.33mol/L KCL aqueous solution diffusing at 298.15K, demonstrating that the designed DLCL works well in measuring liquid diffusion coefficients.

2. Design scheme and theoretical analysis

The accurate measurement of the D value is mainly determined by the precise measurement of the RI of liquid filled in the diffusion pool. RI sensitivity, depth of field, and SA are the main factors influencing the measurement. In this section, the designed symmetric liquid-core cylindrical lens and DLCL are introduced in detail.

2.1 Symmetric liquid-core cylindrical lens

Functioned as both diffusion pool and important imaging element, a symmetric liquid-core cylindrical lens with high RI sensitivity is designed firstly. The top view of the symmetric liquid-core cylindrical lens is shown in Fig. 1. Consisting of two identical cylindrical lenses, the geometrical parameters of the symmetric liquid-core cylindrical lens are R₁ = |R₄| = 45.0mm, R₂ = |R₃| = 27.9mm, the thickness of the lens, d₁ = d₄ = 4.0mm, d₂ = d₃ = 3.0mm, length L = 50.0mm, and the material of the symmetric liquid-core cylindrical lens is K9 glass (n₀ = 1.5163 at λ = 589nm). Let the RI of liquid filled in the symmetric liquid-core cylindrical lens be n, O_i be the surface apex, S_i'(O_i) be the distance starting from point O_i, according to the paraxial imaging Gaussian theory, the focal length of symmetric liquid-core cylindrical lens is the function of RI of filled liquid, which can be represented as

Fig. 1 Structure diagram of the designed symmetric liquid-core cylindrical lens. The depth of field is defined as the transversal distance that corresponds to the longitudinal size of a pixel of the CMOS used, which is positioned on the focal plane of the symmetric liquid-core cylindrical lens.

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f (n) = \frac{R_{4} (S_{3}^{'} (O_{3}) - d_{3})}{(n_{0} - 1) (S_{3}^{'} (O_{3}) - d_{3}) + n_{0} R_{4}} + d_{3} + d_{4},

S_{3}^{'} (O_{3}) = \frac{n_{0} R_{3} (S_{2}^{'} (O_{2}) - d_{2})}{(n - n_{0}) (S_{2}^{'} (O_{2}) - d_{2}) + n R_{3}},

S_{2}^{'} (O_{2}) = \frac{n R_{2} (S_{1}^{'} (O_{1}) - d_{1})}{(n - n_{0}) (S_{1}^{'} (O_{1}) - d_{1}) + n_{0} R_{2}},

S_{1}^{'} (O_{1}) = \frac{n_{0} R_{1}}{n_{0} - 1} .

In order to determine the RI of liquid (n) filled in liquid-core cylindrical lens, the image system should be calibrated in advance. First of all, the RI of standard liquid sample was determined by an Abbe Refractometer. When the liquid-core cylindrical lens is filled with a standard liquid sample of RI = n₁ and the CMOS fixed in the translation stage is precisely set on the focal plane of the image system, let S₁ be the reading value on the stage, f₁ (n₁) be the calculation value based on Eq. (1), and calibration value of the image system S₀ = S₁- f₁ (n₁). The calibrated focal length is then f (n) = S-S₀, where S is the reading value when the front liquid core of the DLCL is filled with liquid to be measured, of which RI = n can be known by substituting f (n) = S-S₀ into Eq. (1) [15,18].

If the parameters (R_i, d_i, n₀) of the symmetric liquid-core cylindrical lens are given, the focal length f is a unitary function of the RI (n) that can be deduced from Eq. (1). The RI sensitivity is defined as the change of focal length (Δf) caused by the change of RI (Δn), which is set as 0.0002 in this paper for the convenience of comparison with a normal Abbe refractometer. When collimated light (wavelength λ = 589.0 nm, width 2h = 17.6 mm) is incident on the symmetric liquid-core cylindrical lens filled with pure water,the focal length of the symmetric liquid-core cylindrical lens is f (n = 1.3330) = 104.122 mm, which is longer than the focal length of the symmetric liquid-core cylindrical lens filled with most other normal liquids. The diffractive limit of the symmetric liquid-core cylindrical lens is DL = 1.22λf /2h = 4.3 μm, a value smaller than the pixel size of the CMOS used (σ = 5.5μm). Therefore, the diffraction effect can be neglected, and the measurement error of focal length (δf) depends mainly on the geometrical depth of field [17,18]. As shown in Fig. 1, the relationship between δf and depth of field can be expressed as

δ f = Depth Of Field= \frac{σ}{\tan θ} \approx \frac{σ f}{h} .

If Δf is greater than δf, the resolvable minimum RI is better than the preset Δn = 0.0002, otherwise, it is poorer than 0.0002. The resolvable minimum RI(δn)of the designed symmetric liquid-core cylindrical lens can be expressed as

δ n = {\frac{Δ n}{(Δ f / δ f)} |}_{Δ n = 0.0002} .

For the normal liquid RI (n = 1.3300~1.6000), the curves of Δf, δf, and δn varied with n values have been calculated and shown in Fig. 2 by red, black and green lines, respectively. It is clear that when the RI of liquid core is less than 1.5170 marked by dotted arrow in Fig. 2, Δf is larger than δf, and the resolvable minimum RI (δn) is better than 0.0002; otherwise, δn is poorer than 0.0002. Some values of widely used liquids are also marked in Fig. 2, which indicates that the δn is 8.8*10⁻⁵, 10.5*10⁻⁵, 14.8*10⁻⁵, 17.4*10⁻⁵ and 22.0*10⁻⁵, for pure water, alcohol, ethylene glycol (EG), glycerin and nitrobenzene, respectively.

Fig. 2 Characteristic curves of the designed symmetric liquid-core cylindrical lens. The red line is the change of focal length (Δf) caused by the change of liquid RI (Δn = 0.0002); the dark line is the measurement error of focal length (δf); the green line is the resolvable minimum RI (δn). Water (n = 1.3330), alcohol (n = 1.3610), EG (n = 1.4310), glycerol (n = 1.4730), and nitrobenzene (n = 1.5500) are marked on the curves. The cross point between the curves of Δf and δf corresponds to the RI (n = 1.5170), of which resolvable minimum RI is δn = 0.0002 exactly.

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Aberration is another important factor considered in the design of a liquid-core cylindrical lens, SA is the main aberration for a cylindrical symmetry lens when a monochromic laser (λ = 589.0 nm) is illumination light, which spoils the image quality and interferes with judgment of focal position. The SA of the designed symmetric liquid-core cylindrical lens on the image plane has been simulated by using an optical analysis software (ZEMAX). When the RIs of liquid filled in the core of symmetric liquid-core cylindrical lens are n = 1.3330, 1.3350 and 1.4310, respectively, the focus spot diagrams are shown in Figs. 3(a1)-3(a3); while the SA curves are shown in Figs. 3(b1)-3(b3). Figures 3(a2) and 3(b2) indicate that SA is almost zero at the focal position of the designed symmetric liquid-core cylindrical lens as n = 1.3350, however, the SA is large either in Figs. 3(a1) and 3(b1) as n = 1.3330 or in Figs. 3(a3) and 3(b3) as n = 1.4310. Figure 3 demonstrates that it is impossible to eliminate SA of a symmetric liquid-core cylindrical lens simultaneously when its core is filled with different liquids.

Fig. 3 SA simulation results of the designed symmetric liquid-core cylindrical lens when the width of the incident light (entrance pupil) is 17.6 mm. (a_i, i = 1,2,3) is the focus spot diagram, and (b_i i = 1,2,3) is SA diagram. The liquid core of the symmetric liquid-core cylindrical lens are filled with pure water (i = 1, n = 1.3330), mixed liquid (i = 2, n = 1.3350) and EG (i = 3, n = 1.4310), respectively. The abscissa in SA diagram is normalized by the entrance pupil.

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2.2 Double liquid-core cylindrical lens (DLCL)

In order to design a liquid-core cylindrical lens that keeps an good RI measurement accuracy, at the same time, the SA produced by the liquid-core cylindrical lens can be eliminated easily, no matter what kind of liquids are filled in the core of the liquid-core cylindrical lens, an additional cylindrical lens is set behind the liquid-core cylindrical lens, which makes up the DLCL when the space between the liquid-core cylindrical lens and the added lens is filled with required liquid. The top view of the DLCL is shown in Fig. 4. The geometrical parameters are R₁ = |R₄| = 45.0 mm, R₂ = |R₃| = 27.9 mm, R₅ = 21.5 mm, R₆ = ∞; the thickness of the lens are d₁ = d₄ = 4.0 mm, d₂ = d₃ = 3.0 mm, d₅ = 1.0 mm, d₆ = 12.0 mm, length L = 50.0 mm; the material of the symmetric liquid-core cylindrical lens is K9 glass (n₀ = 1.5163 at λ = 589.0 nm), the RIs of liquids filled in the front and rear liquid cores are n and n’ , respectively.

Fig. 4 Structure diagram of the designed DLCL.

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Based on the paraxial imaging Gaussian theory, the focal length of DLCL is the function of both n and n’, which can be represented as

f (n, n^{'}) = \frac{S_{5}^{'} (O_{5}) - d_{6}}{n_{0}} + d_{3} + d_{4} + d_{5} + d_{6},

S_{5}^{'} (O_{5}) = \frac{n_{0} R_{5} (S_{4}^{'} (O_{4}) - d_{5})}{n^{'} R_{5} + (n_{0} - n^{'}) (S_{4}^{'} (O_{4}) - d_{5})},

S_{4}^{'} (O_{4}) = \frac{n^{'} R_{4} (S_{3}^{'} (O_{3}) - d_{4})}{n_{0} R_{4} - (n^{'} - n_{0}) (S_{3}^{'} (O_{3}) - d_{4})},

S_{3}^{'} (O_{3}) = \frac{n_{0} R_{3} (S_{2}^{'} (O_{2}) - d_{2} - d_{3})}{n R_{3} - (n_{0} - n) (S_{2}^{'} (O_{2}) - d_{2} - d_{3})},

S_{2}^{'} (O_{2}) = \frac{n (S_{1}^{'} (O_{1}) - d_{1}) R_{2}}{n_{0} R_{2} + (n - n_{0}) (S_{1}^{'} (O_{1}) - d_{1})},

S_{1}^{'} (O_{1}) = \frac{n_{0} R_{1}}{n_{0} - 1} .

In the range of liquid n = 1.3300~1.6000, the resolvable minimum RI of the DLCL has been analyzed in the same way as that for the symmetric liquid-core cylindrical lens. When the core of the rear lens is filled with EG aqueous solution (n' = 1.4042), the curves of Δf, δf, and δn varied with n values for DLCL have been calculated and shown in Fig. 5. For pure water, the RI sensitivity (Δf) increases from 146.9 (symmetric liquid-core cylindrical lens) to 343.0μm (DLCL), while the resolvable minimum RI (δn) improves from 8.8*10⁻⁵ (symmetric liquid-core cylindrical lens) to 5.8*10⁻⁵ (DLCL). For EG, Δf increases from 51.1to 81.8μm, while the δn improves from 14.8*10⁻⁵ to 11.9*10⁻⁵. In the range of liquid n = 1.3300~1.5646, the resolvable minimum RI is better than 0.0002, which is marked by a dotted line in Fig. 5.

Fig. 5 Characteristic curves of the designed DLCL when liquid of n' = 1.4042 is filled in the rear core. Meaning of the signs is the same as that in Fig. 2.

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In addition to improving resolvable minimum RI, the designed DLCL reduces the SA dramatically by selecting liquid filled in the core of rear lens. For examples, when the liquid filled in the front core is water (n = 1.3330), the selected liquid in the core of rear lens is the mixed solution of n' = 1.3983. The SA curves simulated by the ZEMAX are shown in Figs. 6(a1) and 6(b1), which indicates that the SA is reduced from ± 10 μm (symmetric liquid-core cylindrical lens) to ± 5μm (DLCL) as n = 1.3330 and n' = 1.3983. When the water is replaced by EG (n = 1.4310), the selected liquid is the mixed solution of n' = 1.4493, the Figs. 6(a2) and 6(b2) indicate that the SA is reduced from ± 500 μm (symmetric liquid-core cylindrical lens) to ± 2μm (DLCL).

Fig. 6 SA simulation results of DLCL when the width of the incident light (entrance pupil) is 17.6 mm. (a_i, i = 1,2) is the focus spot diagram and (b_i, i = 1,2) is SA diagram. (a₁) and (b₁): pure water (n = 1.3330) and the liquid of n’ = 1.3983 are filled in the front and rear liquid cores, respectively. (a₂) and (b₂): EG (n = 1.4310) and the liquid of n’ = 1.4493 are filled in the front and rear liquid cores, respectively. The abscissa in SA diagram is normalized by the entrance pupil.

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If the core of rear lens is filled with liquid of n' = 1.3983, 1.4020, 1.4253, 1.4493, 1.4997, and 1.6216, respectively, the curves of SA varied with RI (n) of liquid in the front core have been calculated and shown in Fig. 7 for the designed DLCL, which indicates that SA reaches its minimum for water, alcohol, 70% EG aqueous solution, EG, glycerol, and nitrobenzene.

Fig. 7 The SAs varied with liquid RI (n) for different liquid RI (n') filled in the rear core of DLCL. The dotted arrows indicate the RI positions for different liquids filled in the front core: water (n = 1.3330), alcohol (n = 1.3610), 70%-EG (n = 1.4050), EG (n = 1.4310), glycerol (n = 1.4730) and nitrobenzene (n = 1.5500).

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A diffusion image with a small SA at a specific RI value is important in measuring coefficient of liquid diffusion [15], however, the SA in a range of liquid RI, says n_i to n_f, is also required to be as small as possible in some applied situations [18,19]. For examples, the SA of the designed DLCL is required to be small in the range of n_i = 1.3330 to n_f = 1.4310 for the experiment of EG (n_f) diffusing in water (n_i), the sum of SA varied with the RI in the rear liquid core (n') at the interval of Δn = 0.0002 has been calculated and shown in Fig. 8, which indicates that there is a minimum value as n' = 1.4042. When n' is fixed at 1.4042, the SA is smaller than70 μm in a wide range of n = 1.3330 to 1.4310, that is shown in the insert of Fig. 8.

Fig. 8 The sum of SA in the range of n = 1.3330 to 1.4310 varied with the RI of liquid filled in the rear core of designed DLCL. The dotted arrow indicates the RI position where the sum of SA reaches its minimum. The SA varied with the RI of liquid (n), which is filled in the front core when the RI of liquid in the rear core is fixed at n' = 1.4042, is inserted at the upper right corner.

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In order to verify the effect of DLCL, collimated light beams passing through the designed DLCL were detected by a CMOS camera positioned at the focal plane, and the results were compared with that of corresponding symmetric liquid-core cylindrical lens. Filled with liquid of n' = 1.4042, the images obtained by the DLCL is shown in Fig. 9, where the first column is the images obtained by the symmetric liquid-core cylindrical lens when its core is filled with different EG aqueous solutions; the second column is its intensity profile; the third column is the images obtained by the DLCL and the fourth column is its intensity profile. It is clear by the comparison that, due to SA reducing works, the images obtained by the DLCL are much sharper than that obtained by the symmetric liquid-core cylindrical lens in the range of n = 1.3330 to 1.4310.

Fig. 9 Focus images and light intensity profiles. (a) and (b): H₂O (n = 1.3330), 30%-EG (n = 1.3652), 50%-EG (n = 1.3850), 80%-EG (n = 1.4139) and 100%-EG (n = 1.4310) are filled in the core of symmetric liquid-core cylindrical lens; (c) and (d): the same solutions are filled in the front core of DLCL when the rear core is filled with liquid of n' = 1.4042.

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3. Diffusion experiment of 0.33Mol/L KCL aqueous solution

The imaging principle is shown in Fig. 10. The high concentration of KCl solution (3mol/L) was injected to the front core of DLCL’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 turbulence, and then the 0.33 mol/L KCl solution was introduced to the DLCL’s upper half part along the inner wall of asymmetric liquid-core cylindrical lens by the same digital syringe with a slow speed (0.4ml/ min), making sure no obvious turbulence current in the two liquids. The injection of 0.33 mol/L KCl solution took about 7 minutes, the time of two solutions starting to contact was defined as the onset of diffusion (t = 0). Once the two solutions contact together, the diffusion process commences. Dynamic gradient distributions of concentrations for the mixed solution is formed gradually along the diffusion direction, which is defined as the Z-axis as shown in Fig. 10. The collimated light beams passing through the DLCL will project a dynamic “beam waist” image on the CMOS as shown in the right side of Fig. 10, and there is a sharp point $Z_{c}^{'}$ on the CMOS that contributes to a thin liquid layer of n = n₃ = n_c. The dynamic image reflects the diffusion process, based on the Fick’s second law, the D value can be figured out either by computing the shift rate of the selected thin liquid layer [18] in the diffusion cell or by analyzing an instantaneous diffusion image at a certain moment [15].

Fig. 10 Illustration of the imaging principle for DLCL filled with diffusion liquids in the front of liquid core. A RI gradient distribution of the filled liquid is formed along Z-axis, n₁<n₂ <n₃ = n_c <n₄.

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3.1 The first method to calculate the diffusion coefficient

Let binary solutions involved in diffusion be A and B, C be the concentration of A in B, the initial concentrations be C₁ and C₂ on two sides of interface (Z = 0) before the diffusion beginning. According to the boundary and initial conditions, the solution of Fick's second law [3] is the error function that can be expressed as [1]

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 C(Z, t) is the concentration at diffusion time t and position Z, D is the diffusion coefficient, Z is the distance from the diffusion interface,

e r f (Z / 2 \sqrt{D t})

is the Gauss error function. The position Z can be written as an inverse Gauss error function, i.e.,

Z^{'} = \sqrt{D t} \cdot e r f i n v {[C (Z, t) - \frac{C_{1} + C_{2}}{2}] / (\frac{C_{1} - C_{2}}{2})} - Δ Z .

where △Z is the error caused by measuring the diffusion interface. The relationship between solution concentration and its RI satisfies with C(Z', t) = f [n(Z', t)], which is usually a linear function, C = f(n) = mn + C₀, and can be pre-determined experimentally. In the first method for calculating D value, a thin liquid layer with a fixed concentration, and then a fixed RI, C_c(Z', t) = f [n_c(Z', t)], is chosen to focus clearly on the CMOS plane. Let C₁ = 0.33 Mol/L, C₂ = 3.00 Mol/L, C_c = f(n_c) = 111.77n_c-149.02 (Mol/L, experimentally fitting result, R² = 0.9996), the focal position varied with diffusion time can be rewritten as

Z^{'} = \sqrt{D} \cdot e r f i n v {- [111.77 \times n_{c} - 148.855] / 1.335} \sqrt{t} - Δ Z .

Eq. (7) is used to determine D value by comparing it with experimental result.

Liquid diffusion coefficient, in generally, is dependent on solution concentration, to measure the D valve of aqueous solution of 0.33Mol/L KCL, the concentration of selected thin liquid layer should be close to that of 0.33Mol/L KCL [19]. The RI of 0.33Mol/L KCL is n = 1.3364 at room temperature (298.15K) based on a measurement with an Abbe Refractometer, the selected RI of the thin liquid layer in our experiment is n = n_c = 1.3376, corresponding to 0.48 Mol/L KCL solution. In order to focus collimation beams passing through the thin liquid layer on the CMOS as sharp as possible, the SA varied with the RI (n') in the rear liquid core has been calculated when the RI in the front liquid core fixed at n = 1.3376, the result is shown in Fig. 11, which indicates that the SA for the thin liquid layer of n = 1.3376 reaches its minimum as n' = 1.3989. When n' is fixed at 1.3989, the SA is smaller than 2 μm in a wide range of n = 1.3364 to 1.3460 that is shown in the insert of Fig. 11. Based on above analysis, the RI of liquid filled in the rear liquid core of DLCL is n' = 1.3989.

Fig. 11 The SA varied with the RI of liquid filled in the rear core of designed DLCL. The dotted arrow indicates the RI position where the SA reaches its minimum. The SA varied with the RI of liquid (n), which is filled in the front core when the RI of liquid in the rear core is fixed at n’ = 1.3989, is inserted at the upper right corner.

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Pure water filled in the front core of DCLC was used to calibrate measurement system. After system calibration, the aqueous solution of 3.00 Mol/L KCL was injected into the bottom of the DLCL’s front core, and then the same volume of 0.33 Mol/L KCL aqueous solution was trickled slowly along the inner wall. The diffusion images were taken by a CMOS camera (4096 pixels × 3072 pixels) at the interval of 120s, some of the images during1200 to 3600s were shown in Fig. 12, and the focal position (Z') varied with diffusion time (t) were listed in the Table 1.

Fig. 12 Diffusion images varied with time.

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Table 1. Data of focal position (Z') varied with diffusion time. A random integer among −8 to 8 is added on the focal position (Z'_rdm) to estimate experimental deviation.

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The Z' and $\sqrt{t}$ in the Table 1 are fitted linearly by the least square method, the fitting result is $Z^{'} = 95.884 \sqrt{t} - 61.025$ in the unit of micrometer (R² = 0.9995), which deduces the diffusion coefficient D = 1.8508 × 10⁻⁵cm²/s from comparing the fitting result with Eq. (7), a value that is very close to the literature value [20, 21] (D_lit = 1.8410 × 10⁻⁵cm²/s) measured by interference method.

3.2 The second method to calculate the diffusion coefficient

In the first method to determine D value, it is required to take many diffusion images at different time, which will expend a long experiment time (∼one hour). In fact, the spatial distribution of liquid concentration C(Z, t) caused by diffusion is implied in each diffusion image, the D value can be fitted out by analyzing any one of the images shown in Fig. 12, that is called instantaneous image analysis method [15] since only 17 millisconds are required to take one diffusion image. The RIs of KCL aqueous solution for 0.33 and 3.00 Mol/L at 298.15 K are n = 1.3364 and 1.3600, respectively, to keep the diffusion images a minimum SA in the range of n = 1.3364 to 1.3600, the sum of SA in the same range varied with the RI in the rear liquid core (n') at the interval of Δn = 0.0002 has been calculated and shown in Fig. 13, a minimum value arrives at n' = 1.3994. When n' is fixed at 1.3994, the SA varied with the RI in the front liquid core (n) is then calculated and shown in the insert of Fig. 13, which indicates that the SA is smaller than 5.5μm in a wide range of n = 1.3364 to 1.3600.

Fig. 13 The sum of SA in the range of n = 1.3364 to 1.3600 varied with the RI of liquid filled in the rear core of designed DLCL. The dotted arrow indicates the RI position where the sum of SA reaches its minimum. The SA varied with the RI of liquid (n), which is filled in the front core when the RI of liquid in the rear core is fixed at n' = 1.3394, is inserted at the upper right corner.

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Based on the analysis, the liquid layer of n = n_c = 1.3376 was selected to image sharply on the CMOS, while the RI in the rear liquid core was fixed at n' = 1.3994. The diffusion images in the period of 1200 to 3600s were taken by an interval of 120s. One of the images taken at 1800s was shown in Fig. 14.

Fig. 14 One of the instantaneous diffusion images taken at t = 1800s. The image widths at different positions (Σ_i, Z_i) are plotted in the figure.

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Assuming Σ the width of diffusion image as shown in Fig. 14, the relationship between liquid concentration and Σ in the unit of pixel is C(Σ) = −0.0037Σ + 0.4835 (Mol/L), which is pre-determined by experimental method. Let f(Σ) = erfinv((0.0037Σ−1.1815)/1.3350), for the image taken at diffusion time t_0, Eq. (7) is expressed as

Z = \sqrt{D t_{0}} f (Σ) - Δ Z .

Eq. (8) is used to determine D value by comparing it with experimental results.

The experimental fitting results for the diffusion images taken from 1200s to 3600s in an interval of 120s are shown in Table 2, the average value is $\bar{D}$ = 1.8619 × 10⁻⁵ cm²/s and the standard deviation is σ = 0.0749 × 10⁻⁵. The large deviation before t = 1680s may be caused by the un-damped interface turbulence, while the deviation after t = 3120s is probably caused by the reading error of the image width.

Table 2. Data of instantaneous image analysis method at different diffusion time. A random integer among (−1, 0, 1) is added on the image width (Σ) to estimate experimental deviation.

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3.3 Error analysis for two measurement methods

For the first method to determine D value, we think that the experimental deviation is mainly caused by the error in reading focal position (Z '), which may occupy several pixels along the diffusion direction (Z-axis), but is of same image width. The phenomenon becomes obvious when the concentration gradient of diffusion solution is small. How many pixels will the focal position occupy and how large deviation will it cause? The function of concentration gradient is used to analyze those problems, which is deduced from Eq. (5) and written as

\frac{d C}{d Z} = - \frac{(C_{1} - C_{2})}{2 \sqrt{π D t}} e^{- {(\frac{Z}{2 \sqrt{D t}})}^{2}} .

For the image taken at t₀ = 3600s (or 1200s), Z_i' = 1028 pixel (590 pixel), C = mn + C₀ = 111.77n −149.02 (Mol/L), n = n_c = 1.3376. Let D = 1.8508 × 10⁻⁵cm²/s, C₁ = 0.33 Mol/L and C₂ = 3.00 Mol/L, the interval (δZ) caused by a RI deviation (Δn) can be written as $δ Z = (- 2 \sqrt{π D t} / (C_{1} - C_{2})) \exp {(Z / 2 \sqrt{D t})}^{2} m Δ n$ . Assuming Δn = δn = 6.11 × 10⁻⁵ at n = n_c = 1.3376, where δn is the minimum resolvable RI computed from Eq. (3), the calculated δZ is 77.7 μm (44.2 μm) corresponding to 15 pixels (9 pixels) along the Z-axis. A random integer between −8 and 8 is then added to the measured Z'_i, which is labeled by Z'_rdm and listed in the 3th and 6th columns of Table 1. The fitting result between Z'_rdm and $\sqrt{t_{i}}$ is $Z_{rdm}^{'} = 96.362 \sqrt{t_{i}} - 82.819$ (μm, R² = 0.9985), leading to D = 1.8692 × 10⁻⁵cm²/s by comparing it with Eq. (7). Therefore, the relative deviation caused by the error in reading focal position is less than 1.0%.

Only one instantaneous diffusion image is required to determine D value in the second method introduced, therefore, the time spent on the experiment is shorten greatly. Because image widths at different positions (Σ(Z_i)) are required to fit experimental data, the SA in a diffusion image has great influence on the measurement result. When a liquid layer of which RI is n_c = 1.3376 is selected to image sharply on the CMOS plane, as shown in Fig. 14, there is an optimized RI in the rear liquid core of the DLCL (n' = 1.3994), which reduces the SA to the minimum. The SA in the data acquiring range (n = 1.3364 to 1.3376), as shown in the insert of Fig. 13, is smaller than 1.6 μm that is limited to one pixel of the CMOS chip used.

To check the influence of SA on the measured D value, an integer among (1, 0, −1) is added randomly on the measured image width in the unit of pixel, the fitting results between Z_rdm and Σ_rdm are listed in the 5th column, the computed D values labeled as D_rdm are listed in the 6th column, and the differences between D and D_rdm are listed in the 7th column of Table 2. The average value is $\bar{D_{rdm}}$ = 1.7559 × 10⁻⁵ cm²/s, the standard deviation is σ = 0.1270 × 10⁻⁵ cm²/s, and the relative deviation between $\bar{D_{rdm}}$ and $\bar{D}$ is 5.7%. It is clear that one-pixel random deviation in measuring image width has a large influence on the calculation of D value, the works on reducing SA is very important for obtaining D correctly, due to the SA is the main cause of image width deviation for the second measurement method.

4. Summary

In order to measure the binary liquid diffusion coefficient with a liquid core cylindrical lens, an aplanatic DLCL has been designed and fabricated successfully. The DLCL is composed by two liquid-core cylindrical lenses, the front lens of the DLCL is used as both diffusion cell and key imaging element, the RI of liquid filled in its core can be measured in the way of spatial resolution; the rear lens of the DLCL is used as an aplanatic component, either the RI position of SA or the SA in a range of RI caused by the front lens can be regulated by selecting the liquid, of which RI is known and filled in the rear liquid core. With the help of the DLCL, the D value of 0.33mol/L KCL aqueous solution at temperature 298.15K has been measured by using two methods, the first method derives D value precisely from the drift rate of diffusion image, the measured value is D = 1.8508 × 10⁻⁵cm²/s; the second method obtains D value rapidly by analyzing an instantaneous diffusion image, the measured value is D = 1.8619 × 10⁻⁵cm²/s. The measured values are in good consistent with the literature value using interference method, D_lit = 1.8410 × 10⁻⁵cm²/s, which is the conventional criterion value. Finally, the errors of two measurement methods are analyzed. We find that experimental deviation is mainly caused by the error in reading focal position for the first method, the relative deviation in measuring D value is less than 1.0%; while the error in measuring image width is the main deviation for the second method, one-pixel random error in reading the image width may lead to a the relative deviation 5.7% in measuring D value. The deviation analysis demonstrates that the works on reducing SA of the DLCL are very important for the method using a liquid core cylindrical lens.

Funding

National Natural Science Foundation of China (NSFC) (Grant NO. 61465014); Applied Basic Research Key Project of Yunnan (Grant NO. 2016CYH05); Scholarship Award for Excellent Doctoral Student granted by Yunnan Province, China.

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. (University of Minnesota, 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. D. Ambrosini, D. Paoletti, and N. Rashidnia, “Overview of diffusion measurements by optical techniques,” Opt. Lasers Eng. 46(12), 852–864 (2008). [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. N. Bochner and J. Pipman, “A simple method of determining diffusion constants by holographic interferometry,” J. Phys. D 9(13), 1825–1830 (1976). [CrossRef]

9. F. Ruiz-Bevia, J. Fernandez-Sempere, A. Celdran-Mallol, and C. Santos-Garcia, “Liquid diffusion measurement by holographic interferometry,” Can. J. Chem. Eng. 63(5), 765–771 (1985). [CrossRef]

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. 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]

12. 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]

13. G. Taylor, “Diffusion and mass transport in tubes,” Proc. Phys. Soc. B 67(12), 857–869 (1954). [CrossRef]

14. G. Taylor, “Conditions under which dispersion of a solute in a stream of solvent can be used to measure molecular diffusion,”. Proceeding of the Royal Society of London Series A, Mathematical and Physical Sciences, 225 (1163), 473(1954). [CrossRef]

15. L. Sun, W. Meng, and X. Pu, “New method to measure liquid diffusivity by analyzing an instantaneous diffusion image,” Opt. Express 23(18), 23155–23166 (2015). [CrossRef] [PubMed]

16. M. Born, and M. A. Dr Phi, FRS, Principles of Optics (Cambridge University, 1999).

17. A. Ghatak, Optics 4rd ed (McGraw-Hill Education, 2009).

18. L. Sun, C. Du, Q. Li, and X. Pu, “Asymmetric liquid-core cylindrical lens used to measure liquid diffusion coefficient,” Appl. Opt. 55(8), 2011–2017 (2016). [CrossRef] [PubMed]

19. L. Sun and X. Pu, “A novel visualization technique for measuring liquid diffusion coefficient based on asymmetric liquid-core cylindrical lens,” Sci. Rep. 6(28264), 28264 (2016). [CrossRef] [PubMed]

20. B. L. J. Gosting, “A Study of the Diffusion of Potassium Chloride in Water at 25°C with the Gouy Interference Method,” J. Am. Chem. Soc. 72(10), 4418–4422 (1950). [CrossRef]

21. J. A. Rard and D. G. Miller, “Mutual Diffusion Coefficients of BaCl₂-H₂O and KCl-H₂O at 25°Cfrom Rayleigh Interferometry,” J. Chem. Eng. Data 25, 211–215 (1980). [CrossRef]

Time t/s	Z'/Pixel	Z'_rdm/Pixel	Time t/s	Z'/Pixel	Z'_rdm/Pixel
1200	590	582	2520	866	862
1320	620	623	2640	889	883
1440	651	646	2760	909	908
1560	678	679	2880	927	922
1680	704	711	3000	946	951
1800	728	721	3120	965	963
1920	750	758	3240	983	985
2040	776	780	3360	997	1001
2160	801	807	3480	1012	1009
2280	821	826	3600	1028	1035
2400	844	838

Time t/s	Experimental fitting results			One-pixel random deviation added results
Time t/s	Fitting Results /μm	D / ×10⁻⁵ cm²/s	Deviation/ × 10⁻⁵ cm²/s	Fitting Results /μm	D_rdm / × 10⁻⁵ cm²/s	D-D_rdm/ × 10⁻⁵ cm²/s
1200	Z = 3129.8f(Σ)-204.7	2.0408	0.1789	Z_rdm = 3048.1f(Σ)-91.8	1.9357	0.1051
1320	Z = 3217.4f(Σ)-149.8	1.9605	0.0986	Z_rdm = 3081.1f(Σ) + 31.2	1.7980	0.1625
1440	Z = 3347.1f(Σ)-176.9	1.9450	0.0831	Z_rdm = 3183.2f(Σ) + 40.5	1.7592	0.1858
1560	Z = 3481.5f(Σ)-205.8	1.9424	0.0805	Z_rdm = 3522.8f(Σ)-272.3	1.9888	0.0464
1680	Z = 3566.2f(Σ)-139.0	1.8925	0.0306	Z_rdm = 3572.8f(Σ)-151.2	1.8995	0.0070
1800	Z = 3657.5f(Σ)-155.0	1.8582	0.0037	Z_rdm = 3574.9f(Σ)-42.6	1.7750	0.0832
1920	Z = 3771.2f(Σ)-184.0	1.8518	0.0101	Z_rdm = 3649.6f(Σ)-22.5	1.7343	0.1175
2040	Z = 3895.1f(Σ)-345.2	1.8593	0.0026	Z_rdm = 3936.6f(Σ)-399.9	1.8991	0.0398
2160	Z = 4032.6f(Σ)-404.6	1.8822	0.0203	Z_rdm = 3929.7f(Σ)-271.2	1.7873	0.0949
2280	Z = 4018.2f(Σ)-266.8	1.7704	0.0915	Z_rdm = 3836.2f(Σ)-25.3	1.6136	0.1568
2400	Z = 4126.7f(Σ)-273.3	1.7739	0.0880	Z_rdm = 4109.6f(Σ)-257.4	1.7593	0.0146
2520	Z = 4380.6f(Σ)-451.8	1.9037	0.0418	Z_rdm = 4227.8f(Σ)-256.9	1.7732	0.1305
2640	Z = 4449.3f(Σ)-482.8	1.8746	0.0127	Z_rdm = 4225.9f(Σ)-191.4	1.6911	0.1835
2760	Z = 4379.9f(Σ)-298.8	1.7376	0.1243	Z_rdm = 4513.6f(Σ)-479.6	1.8453	0.1077
2880	Z = 4645.2f(Σ)-545.4	1.8731	0.0112	Z_rdm = 4513.4f(Σ)-369.5	1.7683	0.1048
3000	Z = 4635.3f(Σ)-428.1	1.7905	0.0714	Z_rdm = 4519.5f(Σ)-277.6	1.7022	0.0883
3120	Z = 4802.7f(Σ)-546.8	1.8482	0.0137	Z_rdm = 4475.5f(Σ)-121.1	1.6050	0.2432
3240	Z = 4766.2f(Σ)-426.6	1.7528	0.1091	Z_rdm = 4447.6f(Σ)-20.8	1.5263	0.2265
3360	Z = 5025.4f(Σ)-630.9	1.8790	0.0171	Z_rdm = 5009.5f(Σ)-615.5	1.8672	0.0118
3480	Z = 4998.3f(Σ)-521.8	1.7947	0.0672	Z_rdm = 4652.3f(Σ)-73.4	1.5549	0.2398
3600	Z = 5186.4f(Σ)-704.3	1.8680	0.0061	Z_rdm = 4786.0f(Σ)-185.0	1.5906	0.2774

Time t/s	Z'/Pixel	Z'_rdm/Pixel	Time t/s	Z'/Pixel	Z'_rdm/Pixel
1200	590	582	2520	866	862
1320	620	623	2640	889	883
1440	651	646	2760	909	908
1560	678	679	2880	927	922
1680	704	711	3000	946	951
1800	728	721	3120	965	963
1920	750	758	3240	983	985
2040	776	780	3360	997	1001
2160	801	807	3480	1012	1009
2280	821	826	3600	1028	1035
2400	844	838

Time t/s	Experimental fitting results			One-pixel random deviation added results
Time t/s	Fitting Results /μm	D / ×10⁻⁵ cm²/s	Deviation/ × 10⁻⁵ cm²/s	Fitting Results /μm	D_rdm / × 10⁻⁵ cm²/s	D-D_rdm/ × 10⁻⁵ cm²/s
1200	Z = 3129.8f(Σ)-204.7	2.0408	0.1789	Z_rdm = 3048.1f(Σ)-91.8	1.9357	0.1051
1320	Z = 3217.4f(Σ)-149.8	1.9605	0.0986	Z_rdm = 3081.1f(Σ) + 31.2	1.7980	0.1625
1440	Z = 3347.1f(Σ)-176.9	1.9450	0.0831	Z_rdm = 3183.2f(Σ) + 40.5	1.7592	0.1858
1560	Z = 3481.5f(Σ)-205.8	1.9424	0.0805	Z_rdm = 3522.8f(Σ)-272.3	1.9888	0.0464
1680	Z = 3566.2f(Σ)-139.0	1.8925	0.0306	Z_rdm = 3572.8f(Σ)-151.2	1.8995	0.0070
1800	Z = 3657.5f(Σ)-155.0	1.8582	0.0037	Z_rdm = 3574.9f(Σ)-42.6	1.7750	0.0832
1920	Z = 3771.2f(Σ)-184.0	1.8518	0.0101	Z_rdm = 3649.6f(Σ)-22.5	1.7343	0.1175
2040	Z = 3895.1f(Σ)-345.2	1.8593	0.0026	Z_rdm = 3936.6f(Σ)-399.9	1.8991	0.0398
2160	Z = 4032.6f(Σ)-404.6	1.8822	0.0203	Z_rdm = 3929.7f(Σ)-271.2	1.7873	0.0949
2280	Z = 4018.2f(Σ)-266.8	1.7704	0.0915	Z_rdm = 3836.2f(Σ)-25.3	1.6136	0.1568
2400	Z = 4126.7f(Σ)-273.3	1.7739	0.0880	Z_rdm = 4109.6f(Σ)-257.4	1.7593	0.0146
2520	Z = 4380.6f(Σ)-451.8	1.9037	0.0418	Z_rdm = 4227.8f(Σ)-256.9	1.7732	0.1305
2640	Z = 4449.3f(Σ)-482.8	1.8746	0.0127	Z_rdm = 4225.9f(Σ)-191.4	1.6911	0.1835
2760	Z = 4379.9f(Σ)-298.8	1.7376	0.1243	Z_rdm = 4513.6f(Σ)-479.6	1.8453	0.1077
2880	Z = 4645.2f(Σ)-545.4	1.8731	0.0112	Z_rdm = 4513.4f(Σ)-369.5	1.7683	0.1048
3000	Z = 4635.3f(Σ)-428.1	1.7905	0.0714	Z_rdm = 4519.5f(Σ)-277.6	1.7022	0.0883
3120	Z = 4802.7f(Σ)-546.8	1.8482	0.0137	Z_rdm = 4475.5f(Σ)-121.1	1.6050	0.2432
3240	Z = 4766.2f(Σ)-426.6	1.7528	0.1091	Z_rdm = 4447.6f(Σ)-20.8	1.5263	0.2265
3360	Z = 5025.4f(Σ)-630.9	1.8790	0.0171	Z_rdm = 5009.5f(Σ)-615.5	1.8672	0.0118
3480	Z = 4998.3f(Σ)-521.8	1.7947	0.0672	Z_rdm = 4652.3f(Σ)-73.4	1.5549	0.2398
3600	Z = 5186.4f(Σ)-704.3	1.8680	0.0061	Z_rdm = 4786.0f(Σ)-185.0	1.5906	0.2774

Double liquid-core cylindrical lens utilized to measure liquid diffusion coefficient

Abstract

1. Introduction

2. Design scheme and theoretical analysis

2.1 Symmetric liquid-core cylindrical lens

2.2 Double liquid-core cylindrical lens (DLCL)

3. Diffusion experiment of 0.33Mol/L KCL aqueous solution

3.1 The first method to calculate the diffusion coefficient

3.2 The second method to calculate the diffusion coefficient

3.3 Error analysis for two measurement methods

4. Summary

Funding

References and links

Cited By

Figures (14)

Tables (2)

Equations (17)

Optics Express