Abstract

Real-time digital holography is used to study the diffusion process in transparent liquid solutions. Holograms of an object diffusively reflecting through an experimental cell containing diffusing solutions are recorded at different time instances. The recording medium is a CCD chip. The holographic interference of the object at two time instances is carried out numerically in a PC and is used to determine the diffusion coefficient. Holographic interferometric fringes can be displayed on a PC monitor in near real time. The software developed for this method determines the diffusion coefficients automatically. The calculated diffusion coefficients obtained with this method matched well with literature values.

© 2006 Optical Society of America

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References

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  1. E.-L. Cussler, Diffusion-Mass Transfer in Fluid Systems (Cambridge U. Press, 1997).
  2. W. Jost, Diffusion in Solids, Liquids, and Gases (Academic, 1969).
  3. N. Bochner and J. Pipman, "A simple method of determining diffusion coefficients by holographic interferometry," J. Phys. D: Appl. Phys. 9, 1825-1830 (1976).
    [CrossRef]
  4. L. Gabelmann-Gary and H. Fenichel, "Holographic interferometric study of liquid diffusion," Appl. Opt. 18, 343-345 (1979).
    [CrossRef]
  5. J. Szydlowska and B. Janowska, "Holographic measurement of diffusion coefficients," J. Phys. D: Appl. Phys. 15, 1385-1393 (1982).
    [CrossRef]
  6. F. Ruiz-Bevia, A. Celdran-Mallol, C., Santos-Garcia, and J. Fernandez-Sempere, "Holographic interferometric study of diffusion: a new mathematical treatment," Appl. Opt. 24, 1481-1484 (1985).
    [CrossRef] [PubMed]
  7. J. A. Rard and D. G. Miller, "Mutual diffusion coefficients of BaCl2-H2O and KCl2-H2O at 25 °C from Rayleigh interferometry," J. Chem. Eng. Data 14, 124-126 (1991).
  8. D. Paoletti, G. Schirripa Spagnolo, V. Baigini, and M. Santarsiero, "A new method for measuring diffusivity of liquid binary mixtures using DSPI," Pure Appl. Opt. 2, 489-498 (1993).
    [CrossRef]
  9. G. Schirripa Spagnolo, D. Ambrosini, A. Ponticiello, and D. Paoletti, "A simple method of determining diffusion coefficient by digital laser speckle correlation," J. Phys. III 6, 1117-1125 (1996).
    [CrossRef]
  10. E. E. Alanis, G. C. Romero, and C. C. Martinez, "Interferometric measurement of diffusion coefficients through a scanning laser beam," Opt. Eng. 39, 744-750 (2000).
    [CrossRef]
  11. N. Rashidnia and R. Balasubramaniam, "Development of an interferometer for measurement of the diffusion coefficient of miscible liquids," Appl. Opt. 41, 1337-1342 (2002).
    [CrossRef] [PubMed]
  12. A. Anand, V. K. Chhaniwal, S. Mukherjee, and C. S. Narayanamurthy, "Diffusion studies in liquids by multiple beam interferometer," Opt. Laser Technol. 34, 45-49 (2002).
    [CrossRef]
  13. V. K. Chhaniwal, A. Anand, S. Girhe, D. Patil, N. Subhramanyam, and C. S. Narayanamurthy, "New optical techniques for diffusion studies in transparent liquid solution," J. Opt. A: Pure Appl. Opt. 5, S329-S337 (2003).
    [CrossRef]
  14. G. Schirripa Spagnolo, D. Ambrosini, and D. Paoletti, "Liquid diffusion coefficients by digital moiré," Opt. Eng. 43, 798-805 (2004).
    [CrossRef]
  15. U. Schnars and W. Juptner, "Direct recording of holograms by a CCD-target and numerical reconstruction," Appl. Opt. 33, 179-181 (1994).
    [CrossRef] [PubMed]
  16. U. Schnars and W. Juptner, "Digital recording and reconstruction of holograms in hologram interferometry and shearography," Appl. Opt. 33, 4373-4377 (1994).
    [CrossRef] [PubMed]
  17. U. Schnars, "Direct phase determination in hologram interferometry with use of digitally recorded holograms," J. Opt. Soc. Am. A 11, 2011-2015 (1994).
    [CrossRef]
  18. U. Schnars and W. Juptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
    [CrossRef]
  19. T. Kreis, Holographic Interferometry (Academic, 1996).
  20. J. Crank, The Mathematics of Diffusion(Oxford U. Press, 1972).
  21. M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972).
  22. A. Anand, "Tracing of interference fringes using average gray value and simultaneous row and column scan," Opt. Laser Technol. 35, 73-79 (2003).
    [CrossRef]
  23. J. D. Hatfield, O. W. Edwards, and R. L. Dunn, "Diffusion coefficients of aqeous solutions of ammonium and potassium orthophosphates at 25 °C," J. Phys. Chem. 70, 2555-2561 (1966).
    [CrossRef]

2004 (1)

G. Schirripa Spagnolo, D. Ambrosini, and D. Paoletti, "Liquid diffusion coefficients by digital moiré," Opt. Eng. 43, 798-805 (2004).
[CrossRef]

2003 (2)

A. Anand, "Tracing of interference fringes using average gray value and simultaneous row and column scan," Opt. Laser Technol. 35, 73-79 (2003).
[CrossRef]

V. K. Chhaniwal, A. Anand, S. Girhe, D. Patil, N. Subhramanyam, and C. S. Narayanamurthy, "New optical techniques for diffusion studies in transparent liquid solution," J. Opt. A: Pure Appl. Opt. 5, S329-S337 (2003).
[CrossRef]

2002 (3)

A. Anand, V. K. Chhaniwal, S. Mukherjee, and C. S. Narayanamurthy, "Diffusion studies in liquids by multiple beam interferometer," Opt. Laser Technol. 34, 45-49 (2002).
[CrossRef]

U. Schnars and W. Juptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

N. Rashidnia and R. Balasubramaniam, "Development of an interferometer for measurement of the diffusion coefficient of miscible liquids," Appl. Opt. 41, 1337-1342 (2002).
[CrossRef] [PubMed]

2000 (1)

E. E. Alanis, G. C. Romero, and C. C. Martinez, "Interferometric measurement of diffusion coefficients through a scanning laser beam," Opt. Eng. 39, 744-750 (2000).
[CrossRef]

1996 (1)

G. Schirripa Spagnolo, D. Ambrosini, A. Ponticiello, and D. Paoletti, "A simple method of determining diffusion coefficient by digital laser speckle correlation," J. Phys. III 6, 1117-1125 (1996).
[CrossRef]

1994 (3)

1993 (1)

D. Paoletti, G. Schirripa Spagnolo, V. Baigini, and M. Santarsiero, "A new method for measuring diffusivity of liquid binary mixtures using DSPI," Pure Appl. Opt. 2, 489-498 (1993).
[CrossRef]

1991 (1)

J. A. Rard and D. G. Miller, "Mutual diffusion coefficients of BaCl2-H2O and KCl2-H2O at 25 °C from Rayleigh interferometry," J. Chem. Eng. Data 14, 124-126 (1991).

1985 (1)

1982 (1)

J. Szydlowska and B. Janowska, "Holographic measurement of diffusion coefficients," J. Phys. D: Appl. Phys. 15, 1385-1393 (1982).
[CrossRef]

1979 (1)

1976 (1)

N. Bochner and J. Pipman, "A simple method of determining diffusion coefficients by holographic interferometry," J. Phys. D: Appl. Phys. 9, 1825-1830 (1976).
[CrossRef]

1966 (1)

J. D. Hatfield, O. W. Edwards, and R. L. Dunn, "Diffusion coefficients of aqeous solutions of ammonium and potassium orthophosphates at 25 °C," J. Phys. Chem. 70, 2555-2561 (1966).
[CrossRef]

Abramovitz, M.

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972).

Alanis, E. E.

E. E. Alanis, G. C. Romero, and C. C. Martinez, "Interferometric measurement of diffusion coefficients through a scanning laser beam," Opt. Eng. 39, 744-750 (2000).
[CrossRef]

Ambrosini, D.

G. Schirripa Spagnolo, D. Ambrosini, and D. Paoletti, "Liquid diffusion coefficients by digital moiré," Opt. Eng. 43, 798-805 (2004).
[CrossRef]

G. Schirripa Spagnolo, D. Ambrosini, A. Ponticiello, and D. Paoletti, "A simple method of determining diffusion coefficient by digital laser speckle correlation," J. Phys. III 6, 1117-1125 (1996).
[CrossRef]

Anand, A.

V. K. Chhaniwal, A. Anand, S. Girhe, D. Patil, N. Subhramanyam, and C. S. Narayanamurthy, "New optical techniques for diffusion studies in transparent liquid solution," J. Opt. A: Pure Appl. Opt. 5, S329-S337 (2003).
[CrossRef]

A. Anand, "Tracing of interference fringes using average gray value and simultaneous row and column scan," Opt. Laser Technol. 35, 73-79 (2003).
[CrossRef]

A. Anand, V. K. Chhaniwal, S. Mukherjee, and C. S. Narayanamurthy, "Diffusion studies in liquids by multiple beam interferometer," Opt. Laser Technol. 34, 45-49 (2002).
[CrossRef]

Baigini, V.

D. Paoletti, G. Schirripa Spagnolo, V. Baigini, and M. Santarsiero, "A new method for measuring diffusivity of liquid binary mixtures using DSPI," Pure Appl. Opt. 2, 489-498 (1993).
[CrossRef]

Balasubramaniam, R.

Bochner, N.

N. Bochner and J. Pipman, "A simple method of determining diffusion coefficients by holographic interferometry," J. Phys. D: Appl. Phys. 9, 1825-1830 (1976).
[CrossRef]

Celdran-Mallol, A.

Chhaniwal, V. K.

V. K. Chhaniwal, A. Anand, S. Girhe, D. Patil, N. Subhramanyam, and C. S. Narayanamurthy, "New optical techniques for diffusion studies in transparent liquid solution," J. Opt. A: Pure Appl. Opt. 5, S329-S337 (2003).
[CrossRef]

A. Anand, V. K. Chhaniwal, S. Mukherjee, and C. S. Narayanamurthy, "Diffusion studies in liquids by multiple beam interferometer," Opt. Laser Technol. 34, 45-49 (2002).
[CrossRef]

Crank, J.

J. Crank, The Mathematics of Diffusion(Oxford U. Press, 1972).

Cussler, E.-L.

E.-L. Cussler, Diffusion-Mass Transfer in Fluid Systems (Cambridge U. Press, 1997).

Dunn, R. L.

J. D. Hatfield, O. W. Edwards, and R. L. Dunn, "Diffusion coefficients of aqeous solutions of ammonium and potassium orthophosphates at 25 °C," J. Phys. Chem. 70, 2555-2561 (1966).
[CrossRef]

Edwards, O. W.

J. D. Hatfield, O. W. Edwards, and R. L. Dunn, "Diffusion coefficients of aqeous solutions of ammonium and potassium orthophosphates at 25 °C," J. Phys. Chem. 70, 2555-2561 (1966).
[CrossRef]

Fenichel, H.

Fernandez-Sempere, J.

Gabelmann-Gary, L.

Girhe, S.

V. K. Chhaniwal, A. Anand, S. Girhe, D. Patil, N. Subhramanyam, and C. S. Narayanamurthy, "New optical techniques for diffusion studies in transparent liquid solution," J. Opt. A: Pure Appl. Opt. 5, S329-S337 (2003).
[CrossRef]

Hatfield, J. D.

J. D. Hatfield, O. W. Edwards, and R. L. Dunn, "Diffusion coefficients of aqeous solutions of ammonium and potassium orthophosphates at 25 °C," J. Phys. Chem. 70, 2555-2561 (1966).
[CrossRef]

Janowska, B.

J. Szydlowska and B. Janowska, "Holographic measurement of diffusion coefficients," J. Phys. D: Appl. Phys. 15, 1385-1393 (1982).
[CrossRef]

Jost, W.

W. Jost, Diffusion in Solids, Liquids, and Gases (Academic, 1969).

Juptner, W.

Kreis, T.

T. Kreis, Holographic Interferometry (Academic, 1996).

Martinez, C. C.

E. E. Alanis, G. C. Romero, and C. C. Martinez, "Interferometric measurement of diffusion coefficients through a scanning laser beam," Opt. Eng. 39, 744-750 (2000).
[CrossRef]

Miller, D. G.

J. A. Rard and D. G. Miller, "Mutual diffusion coefficients of BaCl2-H2O and KCl2-H2O at 25 °C from Rayleigh interferometry," J. Chem. Eng. Data 14, 124-126 (1991).

Mukherjee, S.

A. Anand, V. K. Chhaniwal, S. Mukherjee, and C. S. Narayanamurthy, "Diffusion studies in liquids by multiple beam interferometer," Opt. Laser Technol. 34, 45-49 (2002).
[CrossRef]

Narayanamurthy, C. S.

V. K. Chhaniwal, A. Anand, S. Girhe, D. Patil, N. Subhramanyam, and C. S. Narayanamurthy, "New optical techniques for diffusion studies in transparent liquid solution," J. Opt. A: Pure Appl. Opt. 5, S329-S337 (2003).
[CrossRef]

A. Anand, V. K. Chhaniwal, S. Mukherjee, and C. S. Narayanamurthy, "Diffusion studies in liquids by multiple beam interferometer," Opt. Laser Technol. 34, 45-49 (2002).
[CrossRef]

Paoletti, D.

G. Schirripa Spagnolo, D. Ambrosini, and D. Paoletti, "Liquid diffusion coefficients by digital moiré," Opt. Eng. 43, 798-805 (2004).
[CrossRef]

G. Schirripa Spagnolo, D. Ambrosini, A. Ponticiello, and D. Paoletti, "A simple method of determining diffusion coefficient by digital laser speckle correlation," J. Phys. III 6, 1117-1125 (1996).
[CrossRef]

D. Paoletti, G. Schirripa Spagnolo, V. Baigini, and M. Santarsiero, "A new method for measuring diffusivity of liquid binary mixtures using DSPI," Pure Appl. Opt. 2, 489-498 (1993).
[CrossRef]

Patil, D.

V. K. Chhaniwal, A. Anand, S. Girhe, D. Patil, N. Subhramanyam, and C. S. Narayanamurthy, "New optical techniques for diffusion studies in transparent liquid solution," J. Opt. A: Pure Appl. Opt. 5, S329-S337 (2003).
[CrossRef]

Pipman, J.

N. Bochner and J. Pipman, "A simple method of determining diffusion coefficients by holographic interferometry," J. Phys. D: Appl. Phys. 9, 1825-1830 (1976).
[CrossRef]

Ponticiello, A.

G. Schirripa Spagnolo, D. Ambrosini, A. Ponticiello, and D. Paoletti, "A simple method of determining diffusion coefficient by digital laser speckle correlation," J. Phys. III 6, 1117-1125 (1996).
[CrossRef]

Rard, J. A.

J. A. Rard and D. G. Miller, "Mutual diffusion coefficients of BaCl2-H2O and KCl2-H2O at 25 °C from Rayleigh interferometry," J. Chem. Eng. Data 14, 124-126 (1991).

Rashidnia, N.

Romero, G. C.

E. E. Alanis, G. C. Romero, and C. C. Martinez, "Interferometric measurement of diffusion coefficients through a scanning laser beam," Opt. Eng. 39, 744-750 (2000).
[CrossRef]

Ruiz-Bevia, F.

Santarsiero, M.

D. Paoletti, G. Schirripa Spagnolo, V. Baigini, and M. Santarsiero, "A new method for measuring diffusivity of liquid binary mixtures using DSPI," Pure Appl. Opt. 2, 489-498 (1993).
[CrossRef]

Santos-Garcia, C.

Schnars, U.

Spagnolo, G. Schirripa

G. Schirripa Spagnolo, D. Ambrosini, and D. Paoletti, "Liquid diffusion coefficients by digital moiré," Opt. Eng. 43, 798-805 (2004).
[CrossRef]

G. Schirripa Spagnolo, D. Ambrosini, A. Ponticiello, and D. Paoletti, "A simple method of determining diffusion coefficient by digital laser speckle correlation," J. Phys. III 6, 1117-1125 (1996).
[CrossRef]

D. Paoletti, G. Schirripa Spagnolo, V. Baigini, and M. Santarsiero, "A new method for measuring diffusivity of liquid binary mixtures using DSPI," Pure Appl. Opt. 2, 489-498 (1993).
[CrossRef]

Stegun, I. A.

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972).

Subhramanyam, N.

V. K. Chhaniwal, A. Anand, S. Girhe, D. Patil, N. Subhramanyam, and C. S. Narayanamurthy, "New optical techniques for diffusion studies in transparent liquid solution," J. Opt. A: Pure Appl. Opt. 5, S329-S337 (2003).
[CrossRef]

Szydlowska, J.

J. Szydlowska and B. Janowska, "Holographic measurement of diffusion coefficients," J. Phys. D: Appl. Phys. 15, 1385-1393 (1982).
[CrossRef]

Appl. Opt. (5)

J. Chem. Eng. Data (1)

J. A. Rard and D. G. Miller, "Mutual diffusion coefficients of BaCl2-H2O and KCl2-H2O at 25 °C from Rayleigh interferometry," J. Chem. Eng. Data 14, 124-126 (1991).

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

V. K. Chhaniwal, A. Anand, S. Girhe, D. Patil, N. Subhramanyam, and C. S. Narayanamurthy, "New optical techniques for diffusion studies in transparent liquid solution," J. Opt. A: Pure Appl. Opt. 5, S329-S337 (2003).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Phys. Chem. (1)

J. D. Hatfield, O. W. Edwards, and R. L. Dunn, "Diffusion coefficients of aqeous solutions of ammonium and potassium orthophosphates at 25 °C," J. Phys. Chem. 70, 2555-2561 (1966).
[CrossRef]

J. Phys. D: Appl. Phys. (2)

J. Szydlowska and B. Janowska, "Holographic measurement of diffusion coefficients," J. Phys. D: Appl. Phys. 15, 1385-1393 (1982).
[CrossRef]

N. Bochner and J. Pipman, "A simple method of determining diffusion coefficients by holographic interferometry," J. Phys. D: Appl. Phys. 9, 1825-1830 (1976).
[CrossRef]

J. Phys. III (1)

G. Schirripa Spagnolo, D. Ambrosini, A. Ponticiello, and D. Paoletti, "A simple method of determining diffusion coefficient by digital laser speckle correlation," J. Phys. III 6, 1117-1125 (1996).
[CrossRef]

Meas. Sci. Technol. (1)

U. Schnars and W. Juptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

Opt. Eng. (2)

E. E. Alanis, G. C. Romero, and C. C. Martinez, "Interferometric measurement of diffusion coefficients through a scanning laser beam," Opt. Eng. 39, 744-750 (2000).
[CrossRef]

G. Schirripa Spagnolo, D. Ambrosini, and D. Paoletti, "Liquid diffusion coefficients by digital moiré," Opt. Eng. 43, 798-805 (2004).
[CrossRef]

Opt. Laser Technol. (2)

A. Anand, V. K. Chhaniwal, S. Mukherjee, and C. S. Narayanamurthy, "Diffusion studies in liquids by multiple beam interferometer," Opt. Laser Technol. 34, 45-49 (2002).
[CrossRef]

A. Anand, "Tracing of interference fringes using average gray value and simultaneous row and column scan," Opt. Laser Technol. 35, 73-79 (2003).
[CrossRef]

Pure Appl. Opt. (1)

D. Paoletti, G. Schirripa Spagnolo, V. Baigini, and M. Santarsiero, "A new method for measuring diffusivity of liquid binary mixtures using DSPI," Pure Appl. Opt. 2, 489-498 (1993).
[CrossRef]

Other (5)

E.-L. Cussler, Diffusion-Mass Transfer in Fluid Systems (Cambridge U. Press, 1997).

W. Jost, Diffusion in Solids, Liquids, and Gases (Academic, 1969).

T. Kreis, Holographic Interferometry (Academic, 1996).

J. Crank, The Mathematics of Diffusion(Oxford U. Press, 1972).

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972).

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Figures (7)

Fig. 1
Fig. 1

Experimental setup for the determination of diffusion coefficients: L2, second lens; MO, microscopic objective; CL, collimating lens; EL, expanding lens.

Fig. 2
Fig. 2

Coordinate system.

Fig. 3
Fig. 3

Change in refractive index with position.

Fig. 4
Fig. 4

Flow chart of the diffusion coefficient determination process: IFT, inverse Fourier transform.

Fig. 5
Fig. 5

(a) Interference fringes obtained for time instances t 1 = 5100 s and t 2 = 5880 s. (b) Skeltalized phase map. The difference obtained in the extreme points was w = 0.62 cm, yielding a diffusion coefficient of 8.79 × 10−6 cm2∕s.

Fig. 6
Fig. 6

(a) Interference fringes obtained for time instances t 1 = 8460 s and t 2 = 9600 s. (b) Skeltalized phase map. The difference obtained in the extreme points was w = 0.78 cm, yielding a diffusion coefficient of 8.46 × 10−6 cm2∕s.

Fig. 7
Fig. 7

Change in diffusion coefficient with concentration for ammonium dihydrogen phosphate (error bar is 3% of the obtained experimental value).

Tables (1)

Tables Icon

Table 1 Measured Diffusion Coefficient (C1 = 0.4484 M, C2 = 0.5479 M, and Cavg = 0.4981 M) at 25 °C

Equations (13)

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R ( x , y ) = A R ( x , y ) exp [ i φ R ( x , y ) ]
O ( x , y , t 1 ) = A O ( x , y ) exp [ i φ O ( x , y , t 1 ) ] ,
O ( x , y , t 2 ) = A O ( x , y ) exp [ i φ O ( x , y , t 2 ) ] .
I ( x , y ) = 2 A O     2 [ 1 + cos ( Δ φ ) ] .
Δ φ ( x , t 1 , t 2 ) = ( 2 π λ ) L Δ n ( x , t 1 , t 2 ) ,
C ( x , t ) = ( C 1 + C 2 ) 2 + ( C 2 C 1 ) π 0 x / 4 D t exp ( η 2 ) d η .
n ( x , t ) = ( d n d C ) 0 C ( x , t ) + n 0 ,
Δ n ( x , t 1 , t 2 ) = n ( x , t 2 ) n ( x , t 1 ) = ( d n d C ) 0 C 2 C 1 π [ 0 x / 4 D t 2 exp ( η 2 ) d η 0 x / 4 D t 1 exp ( η 2 ) d η ] .
x Δ φ ( x , t 1 , t 2 ) = 0.
D = w 2 [ ( 1 / t 1 ) ( 1 / t 2 ) ] 8 ln ( t 2 / t 1 ) ,
φ ( η , ξ , t 1 ) = arctan Im [ Γ ( η , ξ , t 1 ) ] Re [ Γ ( η , ξ , t 1 ) ] ,
φ ( η , ξ , t 2 ) = arctan Im [ Γ ( η , ξ , t 2 ) ] Re [ Γ ( η , ξ , t 2 ) ] ,
Δ φ ( η , ξ , t 1 , t 2 ) = { φ ( η , ξ , t 1 ) φ ( η , ξ , t 2 ) if   φ ( η , ξ , t 1 ) φ ( η , ξ , t 2 ) φ ( η , ξ , t 1 ) φ ( η , ξ , t 2 ) + 2 π if   φ ( η , ξ , t 1 ) < φ ( η , ξ , t 2 ) .

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