Abstract

We design and fabricate an aplanatic double liquid-core cylindrical lens (DLCL), which is used to measure the binary liquid diffusion coefficient (D). The front lens of the DLCL is used as both a diffusion cell and a key imaging element; the refractive index (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, and either the RI position of spherical aberration (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 pre-prepared 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 at temperature 298.15K. The first method derives D value precisely from the drift rate of diffusion image and the measured D is 1.8508 × 10−5 cm2/s. Meanwhile, the second method obtains the D value rapidly by analyzing an instantaneous diffusion image and the measured D is 1.8619 × 10−5 cm2/s. The measured values are in good agreement with the literature value, demonstrating that the designed DLCL works well in measuring liquid diffusion coefficients.

© 2017 Optical Society of America

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References

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  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]

2016 (2)

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]

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]

2015 (2)

2013 (1)

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)

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]

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]

2008 (1)

D. Ambrosini, D. Paoletti, and N. Rashidnia, “Overview of diffusion measurements by optical techniques,” Opt. Lasers Eng. 46(12), 852–864 (2008).
[Crossref]

2004 (1)

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 (1)

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]

1985 (1)

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]

1980 (1)

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]

1976 (1)

N. Bochner and J. Pipman, “A simple method of determining diffusion constants by holographic interferometry,” J. Phys. D 9(13), 1825–1830 (1976).
[Crossref]

1954 (1)

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

1950 (1)

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]

Ambrosini, D.

D. Ambrosini, D. Paoletti, and N. Rashidnia, “Overview of diffusion measurements by optical techniques,” Opt. Lasers Eng. 46(12), 852–864 (2008).
[Crossref]

Anand, A.

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]

Bochner, N.

N. Bochner and J. Pipman, “A simple method of determining diffusion constants by holographic interferometry,” J. Phys. D 9(13), 1825–1830 (1976).
[Crossref]

Celdran-Mallol, A.

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]

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.

Chhaniwal, V. K.

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]

Di, Z.

Du, C.

Fernandez-Sempere, J.

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]

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]

Gosting, B. L. J.

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]

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.

Li, Q.

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]

Meng, W.

Miller, D. G.

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]

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.

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]

Paoletti, D.

D. Ambrosini, D. Paoletti, and N. Rashidnia, “Overview of diffusion measurements by optical techniques,” Opt. Lasers Eng. 46(12), 852–864 (2008).
[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]

Pipman, J.

N. Bochner and J. Pipman, “A simple method of determining diffusion constants by holographic interferometry,” J. Phys. D 9(13), 1825–1830 (1976).
[Crossref]

Pu, X.

Rard, J. A.

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]

Rashidnia, N.

D. Ambrosini, D. Paoletti, and N. Rashidnia, “Overview of diffusion measurements by optical techniques,” Opt. Lasers Eng. 46(12), 852–864 (2008).
[Crossref]

Ruiz-Bevia, F.

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]

Santos-Garcia, C.

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]

Shen, Z.

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]

Sun, L.

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]

Taylor, G.

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

van Leeuwen, T. G.

Weiss, N.

Zou, Y.

Appl. Opt. (1)

Can. J. Chem. Eng. (1)

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]

J. Am. Chem. Soc. (1)

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]

J. Chem. Eng. Data (1)

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]

J. Chem. Phys. (1)

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]

J. Opt. A, Pure Appl. Opt. (1)

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]

J. Phys. D (1)

N. Bochner and J. Pipman, “A simple method of determining diffusion constants by holographic interferometry,” J. Phys. D 9(13), 1825–1830 (1976).
[Crossref]

J. Phys. D Appl. Phys. (1)

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]

Opt. Express (3)

Opt. Lasers Eng. (1)

D. Ambrosini, D. Paoletti, and N. Rashidnia, “Overview of diffusion measurements by optical techniques,” Opt. Lasers Eng. 46(12), 852–864 (2008).
[Crossref]

Phys. Lett. A (1)

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]

Proc. Phys. Soc. B (1)

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

Sci. Rep. (1)

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]

Other (6)

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]

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

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

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

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

E. L. Cussler, Diffusion Mass Transfer in Fluid systems, 3rd ed. (University of Minnesota, 2009)

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Figures (14)

Fig. 1
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.
Fig. 2
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.
Fig. 3
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. (ai, i = 1,2,3) is the focus spot diagram, and (bi 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.
Fig. 4
Fig. 4 Structure diagram of the designed DLCL.
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.
Fig. 6
Fig. 6 SA simulation results of DLCL when the width of the incident light (entrance pupil) is 17.6 mm. (ai, i = 1,2) is the focus spot diagram and (bi, 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.
Fig. 7
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).
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.
Fig. 9
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.
Fig. 10
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 = nc <n4.
Fig. 11
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.
Fig. 12
Fig. 12 Diffusion images varied with time.
Fig. 13
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.
Fig. 14
Fig. 14 One of the instantaneous diffusion images taken at t = 1800s. The image widths at different positions (Σi, Zi) are plotted in the figure.

Tables (2)

Tables Icon

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.

Tables Icon

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.

Equations (17)

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f( n )= 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 )= 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 )= 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 )= n 0 R 1 n 0 1 .
δf=Depth Of Field= σ tanθ σf h .
δn= Δn ( Δf / δf ) | Δn=0.0002 .
f( n, n )= S 5 ( O 5 ) d 6 n 0 + d 3 + d 4 + d 5 + d 6 ,
S 5 ( O 5 )= 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 )= 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 )= 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 )= 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 )= n 0 R 1 n 0 1 .
C(Z,t)= C 1 + C 2 2 + C 1 C 2 2 erf( Z 2 Dt ).
Z = Dt erfinv{ [ C(Z,t) C 1 + C 2 2 ] / ( C 1 C 2 2 ) }ΔZ.
Z = D erfinv{ [ 111.77× n c 148.855 ]/1.335 } t ΔZ.
Z= D t 0 f(Σ)ΔZ.
dC dZ = ( C 1 C 2 ) 2 πDt e ( Z 2 Dt ) 2 .

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