Abstract

An equation is derived from which the fluorescence obtained from a fluorescent whitener on a given substrate can be calculated from several reflectivity measurements. By use of this equation, one can explain theoretically and quantitatively such effects as the shape of the fluorescence vs concentration curve, the effect of the color of the substrate on fluorescence, the effect of the thickness or opacity of the substrate on fluorescence, and the effect of concentration of whitener on the shape of the emission curve.

© 1964 Optical Society of America

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References

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  1. P. Kubelka and F. Munk, Z. Tech. Physik 11, 593–601 (1931).
  2. P. Kubelka, J. Opt. Soc. Am. 38, 448–457 (1948).
    [Crossref] [PubMed]
  3. T. H. Morton, J. Soc. Dyers Colourists 79, 238–242 (1963).
    [Crossref]
  4. See Ref. 2, Eqs. (19) and (20).
  5. See Ref. 2, Eq. (22).
  6. See Ref. 2, Eq. (28).
  7. D. B. Judd, J. Res. Natl. Bur. Std. 19, 287–317 (1937); Paper Trade J. 106, 39–46 (1938).
    [Crossref]
  8. The ratio α can be calculated from a transmittance curve of the dye in a clear substrate which is chemically similar to the substrate used for dyeing. An example is given in a subsequent paragraph. Alternatively, it may be determined from measurements on the dyed sample by a method to be described in a subsequent publication.
  9. A convenient table for the conversion of R∞ to ρ and back, with a spacing of 0.001 in R∞, is given by D. B. Judd, Color in Business, Science and Industry (John Wiley & Sons, Inc., New York, 1952), pp. 358–362.
  10. P. Kubelka, Ref. 2, Eq. (25a).
  11. In the case of paper, for example, owing to inhomogeneity in the sheet, the measured thickness is not the same as the true effective thickness.
  12. E. Allen, Am. Dyestuff Reptr. 48, 27–29 (1959).
  13. S. N. Glarum and S. E. Penner, Am. Dyestuff Reptr. 43, P310–P314 (1954).
  14. Other examples are given in Ref. 12.
  15. Papermaker’s brightness is defined by TAPPI as the reflectance through a specifically defined filter in a carefully defined instrument. Often the reflectance at 459 mμ is used as an approximation. The reflectance at 440 mμ, used in this study, is probably slightly lower than the true “brightness” value, but is reliable as a relative indication of “brightness.”
  16. E. Allen, J. Opt. Soc. Am. 47, 933–943 (1957); Am. Dyestuff Reptr. 46, 425–432 (1957).
    [Crossref]

1963 (1)

T. H. Morton, J. Soc. Dyers Colourists 79, 238–242 (1963).
[Crossref]

1959 (1)

E. Allen, Am. Dyestuff Reptr. 48, 27–29 (1959).

1957 (1)

1954 (1)

S. N. Glarum and S. E. Penner, Am. Dyestuff Reptr. 43, P310–P314 (1954).

1948 (1)

1937 (1)

D. B. Judd, J. Res. Natl. Bur. Std. 19, 287–317 (1937); Paper Trade J. 106, 39–46 (1938).
[Crossref]

1931 (1)

P. Kubelka and F. Munk, Z. Tech. Physik 11, 593–601 (1931).

Allen, E.

Glarum, S. N.

S. N. Glarum and S. E. Penner, Am. Dyestuff Reptr. 43, P310–P314 (1954).

Judd, D. B.

D. B. Judd, J. Res. Natl. Bur. Std. 19, 287–317 (1937); Paper Trade J. 106, 39–46 (1938).
[Crossref]

A convenient table for the conversion of R∞ to ρ and back, with a spacing of 0.001 in R∞, is given by D. B. Judd, Color in Business, Science and Industry (John Wiley & Sons, Inc., New York, 1952), pp. 358–362.

Kubelka, P.

P. Kubelka, J. Opt. Soc. Am. 38, 448–457 (1948).
[Crossref] [PubMed]

P. Kubelka and F. Munk, Z. Tech. Physik 11, 593–601 (1931).

P. Kubelka, Ref. 2, Eq. (25a).

Morton, T. H.

T. H. Morton, J. Soc. Dyers Colourists 79, 238–242 (1963).
[Crossref]

Munk, F.

P. Kubelka and F. Munk, Z. Tech. Physik 11, 593–601 (1931).

Penner, S. E.

S. N. Glarum and S. E. Penner, Am. Dyestuff Reptr. 43, P310–P314 (1954).

Am. Dyestuff Reptr. (2)

E. Allen, Am. Dyestuff Reptr. 48, 27–29 (1959).

S. N. Glarum and S. E. Penner, Am. Dyestuff Reptr. 43, P310–P314 (1954).

J. Opt. Soc. Am. (2)

J. Res. Natl. Bur. Std. (1)

D. B. Judd, J. Res. Natl. Bur. Std. 19, 287–317 (1937); Paper Trade J. 106, 39–46 (1938).
[Crossref]

J. Soc. Dyers Colourists (1)

T. H. Morton, J. Soc. Dyers Colourists 79, 238–242 (1963).
[Crossref]

Z. Tech. Physik (1)

P. Kubelka and F. Munk, Z. Tech. Physik 11, 593–601 (1931).

Other (9)

Other examples are given in Ref. 12.

Papermaker’s brightness is defined by TAPPI as the reflectance through a specifically defined filter in a carefully defined instrument. Often the reflectance at 459 mμ is used as an approximation. The reflectance at 440 mμ, used in this study, is probably slightly lower than the true “brightness” value, but is reliable as a relative indication of “brightness.”

See Ref. 2, Eqs. (19) and (20).

See Ref. 2, Eq. (22).

See Ref. 2, Eq. (28).

The ratio α can be calculated from a transmittance curve of the dye in a clear substrate which is chemically similar to the substrate used for dyeing. An example is given in a subsequent paragraph. Alternatively, it may be determined from measurements on the dyed sample by a method to be described in a subsequent publication.

A convenient table for the conversion of R∞ to ρ and back, with a spacing of 0.001 in R∞, is given by D. B. Judd, Color in Business, Science and Industry (John Wiley & Sons, Inc., New York, 1952), pp. 358–362.

P. Kubelka, Ref. 2, Eq. (25a).

In the case of paper, for example, owing to inhomogeneity in the sheet, the measured thickness is not the same as the true effective thickness.

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Figures (11)

Fig. 1
Fig. 1

Absorbance curve of typical fluorescent whitener. The scale on the right shows α, the ratio of absorbance at the wavelength in question to the absorbance at 365 mμ.

Fig. 2
Fig. 2

Fluorescence vs ρf curves for whiteners of different α value. The quantity ρf is proportional to concentration. In this figure, as well as in all the succeeding ones, the expression “fluorescence ratio” refers to the quantity F′/(λ/λ′)I0.

Fig. 3
Fig. 3

Curves of Fig. 2 replotted on a semilog basis.

Fig. 4
Fig. 4

Linearity of Kubelka–Munk ratio with concentration of fluorescent whitener.

Fig. 5
Fig. 5

Fluorescence vs concentration curve. The solid line was calculated by theory; the circles represent scaled experimental values.

Fig. 6
Fig. 6

Fluorescence vs ρf curves for different values of the scattering power P.

Fig. 7
Fig. 7

Fluorescence vs ρf curves for whiteners applied to substrates of different reflectivity.

Fig. 8
Fig. 8

Response surface of fluorescence on paper as a function of reflectivity of substrate and amount of whitener exhausted. The dots are experimental points; the corresponding theoretical points are the crosses in the surface.

Fig. 9
Fig. 9

Observed vs calculated fluorescence values for dyeings of fluorescent whitener on substrates predyed with Vat Gray 2G. The dashed line represents theory; the deviation of the experimental line indicates quenching of fluorescence by the Vat Gray 2G.

Fig. 10
Fig. 10

Effect of number of sheets of paper on fluorescence for papers of two different degrees of opacity. The top curve is the more opaque sample. Curves were calculated by theory; circles are experimental scaled fluorescence values obtained on the Beckman DK, and crosses are the same obtained on a fluorimeter.

Fig. 11
Fig. 11

Emission curves for dyeings of 1.50% whitener (dashed line) and 0.01% whitener (solid line), scaled so as to be roughly the same height. The dotted curve is a theoretical calculation of the curve for the 1.50% dyeing from the curve of the 0.01% dyeing.

Tables (1)

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Table I Theoretical and experimental fluorescence values on various paper substrates.

Equations (31)

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- d j x = - ( S + K ) j x d x + S i x d x , d i x = - ( S + K ) i x d x + S j x d x ,
j x = C 1 e b S x + C 2 e - b S x ,
i x = C 1 ( a - b ) e b S x + C 2 ( a + b ) e - b S x ,
a = ( S + K ) / S ;             b = ( K 2 + 2 K S ) 1 2 / S .
i x = C 1 R e b S x + C 2 R - 1 e - b S x .
j x = R I 0 [ e b S ( 2 X - x ) - e b S x ] / ( e 2 b S X - R 2 ) , i x = I 0 [ e b S ( 2 X - x ) - R 2 e b S x ] / ( e 2 b S X - R 2 ) .
( i + j ) x = I 0 ( 1 + R ) × [ e b S ( 2 X - x ) - R e b S x ] / ( e 2 b S X - R 2 ) .
d l x = K f ( λ / λ ) ( i + j ) x d x .
d l x = { K f ( λ / λ ) I 0 ( 1 + R ) × [ e b S ( 2 X - x ) - R e b S x ] / ( e 2 b S X - R 2 ) } d x .
d u x = ( 1 + R b ( x ) ) d l x 2 ( 1 - R t ( x ) R b ( x ) ) .
R = [ ( R g - R ) / R ] - R ( R g - 1 / R ) e S X [ ( 1 / R ) - R ] R g - R - ( R g - 1 / R ) e S X [ ( 1 / R ) - R ] .
R t ( x ) = R ( e 2 b S x - 1 ) / ( e 2 b S x - R 2 ) , R b ( x ) = R [ e 2 b S ( X - x ) - 1 ] / [ e 2 b S ( X - x ) - R 2 ] .
d u x = [ e 2 b S ( X - x ) - R ] [ e 2 b S x - R 2 ] d l x 2 ( 1 - R ) ( e 2 b S X - R 2 ) .
T x = b / ( a sinh b S X + b cosh b S X ) .
T x = ( 1 - R 2 ) e b S x / ( e 2 b S x - R 2 ) .
d F x = T x d u x .
d F x = K f ( λ / λ ) I 0 ( 1 + R ) ( 1 + R ) [ e b S ( 2 X - x ) - R e b S x ] [ e b S ( 2 X - x ) - R e b S x ] d x 2 ( e 2 b S X - R 2 ) ( e 2 b S X - R 2 ) .
F = K f ( λ / λ ) I 0 ( 1 + R ) ( 1 + R ) 2 ( e 2 b S X - R 2 ) ( e 2 b S X - R 2 ) 0 X [ e b S ( 2 X - x ) - R e b S x ] [ e b S ( 2 X - x ) - R e b S x ] d x .
F = ( λ / λ ) I 0 ( 1 + R ) ( 1 + R ) ρ f 2 ( e 2 b P - R 2 ) ( e 2 b v P - R 2 ) 0 P [ e b ( 2 P - p ) - R e b p ] [ e b v ( 2 P - p ) - R e b v p ] d p ,
F ( λ / λ ) I 0 = ( 1 + R ) ( 1 + R ) ρ f 2 ( e 2 b P - R 2 ) ( e 2 b v P - R 2 ) { [ e ( b + b v ) P + R R ) ] [ e ( b + b v ) P - 1 ] b + b v - ( R e b P + R e b v P ) ( e b P - e b v P ) b - b v } .
K / S = ( 1 - R ) 2 / 2 R = ρ ;
K / S v = ( 1 - R ) 2 / 2 R = ρ ;
ρ f = ρ - ρ s = ( K / S ) - ( K s / S ) = [ ( 1 - R ) 2 / 2 R ] - [ ( 1 - R s ) 2 / 2 R s ] ;
ρ f = α ρ f ;
ρ f + ρ s = ρ f + [ ( 1 - R s ) 2 / 2 R s ] = ρ ,
b = ( 1 - R 2 ) / 2 R ,
b = ( 1 - R 2 ) / 2 R .
lim P F ( λ / λ ) I 0 = ( 1 + R ) ( 1 + R ) ρ f 2 ( b + b v ) .
P = 1 b s arc coth 1 - a s R s 0 b s R s 0 ,
b s = ( 1 - R 2 s ) / 2 R s , a s = ( 1 + R 2 s ) / 2 R s .
F = I 0 ( 1 - R ) ( K f / K ) ( λ / λ ) × [ ( 1 + R ) b / 2 ( b + b v ) ] .