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

Licun Sun; Weidong Meng; Xiaoyun Pu

doi:10.1364/OE.23.023155

Abstract

A novel optical method was applied to measure the binary liquid diffusion coefficient (D) quickly. Equipped with an asymmetric liquid-core cylindrical lens (ALCL), the spatially resolving ability of the ALCL in measuring refractive index of liquid was utilized to obtain the gradient distribution of the liquid concentration along diffusive direction. Based on Fick’s second law, the D value was then calculated by analyzing diffusion images. It was worth mentioning that only one instantaneous diffusive image was required to measure D value by the method, reducing the measurement time greatly from several hours in traditional methods to a few seconds. The diffusion coefficients of ethylene glycol diffusing in pure water, at temperatures from 288.15 to 308.15 K, were measured by analyzing instantaneous diffusion images, the results were consistent well with the values measured by using holographic interferometry and Taylor dispersion methods. The method is characterized by faster measurement, direct observation of diffusive process, and easy operation, which provides a new method in measuring diffusion coefficient of liquids rapidly.

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Year
Author
Publication

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

2015 (1)

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]

2014 (2)

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]

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]

2013 (2)

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]

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]

2012 (2)

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]

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]

2011 (1)

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).

2009 (1)

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]

2006 (1)

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]

2004 (2)

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]

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]

2003 (2)

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

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]

2002 (1)

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

1996 (1)

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]

1993 (1)

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]

1953 (1)

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

Anand, A.

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]

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]

Angstmann, C.

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]

Bardow, A.

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]

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).

Bedeaux, D.

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]

Caparanga, A. R.

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]

Capuano, F.

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]

Chan, T. C.

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]

Chen, X.

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]

Chhaniwal, V. K.

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]

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]

Colom-Valiente, J.

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]

Culbertson, C. T.

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

D’Errico, G.

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]

Di, Z.

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]

Fernández-Sempere, J.

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]

Girhe, S.

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]

Hampe, M. J.

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]

Jacobson, S. C.

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

Jamshidi-Ghaleh, K.

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]

Kalkman, J.

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]

Kazdal, T. J.

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]

Kjelstrup, S.

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]

Krüger, P.

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]

Kühl, F.

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]

Lang, S.

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]

Langer, R.

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

Li, M. H.

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]

Li, Q.

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]

Liu, X.

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]

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).

Mansour, N.

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]

Más-Pérez, F.

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]

Michael Ramsey, J.

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

Morriss, G. P.

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]

Narayanamurthy, C. S.

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]

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]

Ortona, O.

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]

Othmer, D. F.

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

Patil, D.

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]

Pu, X.-Y.

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]

Ruiz-Beviá, F.

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]

Schnell, S. K.

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]

Shen, Z.

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]

Simon, J.

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]

Snijder, E. D.

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]

Soriano, A. N.

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]

Subrahmanyam, N.

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]

Suh, K. Y.

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

Tang, W. K.

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]

Tavassoly, M. T.

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]

te Riele, M. J. M.

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]

Thakar, M. S.

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

van Leeuwen, T. G.

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]

van Swaaij, W. P. M.

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]

Versteeg, G. F.

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]

Vitagliano, V.

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]

Vlugt, T. J. H.

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]

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).

Wang, M. H.

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]

Weiss, N.

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]

Yang, R.-F.

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]

Zhai, Y.

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]

Zou, Y.

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]

Appl. Opt. (1)

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Supplementary Material (1)

Name	Description
Visualization 1: MP4 (822 KB)	A dynamic process recorded by CCD when the chemical EG diffusing in pure water.

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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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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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Fig. 3 Schematic of the measurement setup.

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

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Fig. 9 Arrhenius equation fitted by experimental datum

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Tables (4)

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

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

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Table 3 The data of position Z_i varied with diffusion time

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

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Equations (16)

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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} .

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

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

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

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} .

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 .

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

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

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

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

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

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

| 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}} .

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.