A NEW EQUATION FOR THE RATE OF FORMATION OF THE PHOTOGRAPHIC LATENT IMAGE* BY JULIAN M. BLAIR AND PHILIP A. LEIGHTON
Introduction An equation which, based upon theoretical postulates, would correctly express the relation between the rate of formation of a developable image and the time and intensity of exposure on photographic emulsions would be of value for the information it might give regarding the nature of the process forming the latent image. Several equations which express a relationship between these quantities have been presented, and have been excellently summarized by Ross,‘ who has also added one of his own. However, these equations either do not fit the experimental data closely, or are not based on theoretical postulates, or both. The research here reported was undertaken with the view of finding an equation which would overcome these difficulties. In the earlier equations, trouble has been caused by the “reciprocity failure,” or lack of inverse proportionality between intensity a n i time of exposure. In order to avoid this- complication we have, in the present work, considered the effect of time of exposure only, while using a uniform light intensity. The Rate of Formation of the Latent Image in Very Thin Emulsions Hurter and Driffield2 showed that the density of a photographic plate is proportional to the mass of silver per unit area. Accordingly, the density of a plate, when developed under such conditions that all developable silver halide grains have been reached and reduced by the developer, will be a measure of the extent to which the reaction forming the latent image has progressed. Ordinary commercial photographic plates are so opaque, due to the concentration of silver salts, that grains of silver halide on the side of the emulsion facing the incident light receive greater exposure than grains buried deeper in the emulsion.s If after exposure and development the density of such a plate is determined it will not be an accurate measure of the degree of developability a t any layer of the emulsion, but rather a summation of the different degrees of developability at the various levels. I n order to overcome this difficulty, we used in this research a special thin emulsion in which the amount of silver halide was so small that the plates
* Contribution from the department of Physics, University of Colorado, and the department of Chemistry, Stanford University. F. E. Rosa: “The Physics of the Developed Photographic Image,” Eastman Kodak Co. Monograph No. 5 , p. 48 (1924). * F. Hurter and V. C. Driffield: Photography, 1890, August 30. Ross: LOC.cit., p. 40.
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JULIAN M. BLAIR AND PHILIP A. LEIGHTON
were quite transparent before development, and grains at different levels of the emulsion could be exposed very nearly equally. Measurement of the density of the special plates before development showed that the amount of silver halide per unit area was constant along lines parallel to the shorter dimensions of the plate, but along lines parallel to the longer dimensions there was a uniform increase in density from one end of the plate to the other. In order to avoid error from this source, the plates were cut into two longitudinal strips, which were reversed in the plate holder so that whenever the dense portion of a strip was given a certain exposure the correspondingly less dense portion of the other strip was given the same exposure. The average density of the two was accepted as the true value of the density. In order to expose the strips, tungsten lamps burning a t constant voltage, were placed behind a ground glass in a ventilated box. On the opposite side of the ground glass an opening in the box was covered with a No. so -Wratten laboratory filter. This filter transmits a band between 4000 and 5000 A, with a maximum transmission around 4550 A. Two and one-half meters in front of the box an ordinary camera plate holder was rigidly mounted in this beam of uniform blue light. As stated, the strips were mounted with ends reversed in this plateholder. The slide of the plateholder was then drawn so that one half inch of the ends of the strips were exposed. After 1 5 seconds the slide was drawn an additional half inch. This was repeated until the whole of the plate was exposed, each half inch of plate receiving I 5 seconds more exposure than the next. On some plates, part of the exposures were of still greater duration, so that the whole range of sensitivity of the plates was explored. Different plates were found to be coated with slightly different amounts of silver halide. Error from this source was avoided by making the greatest exposure on one pair of strips and the least exposure on the next pair of strips. This permitted the densities of all plates to be determined in terms of the amount of silver halide ob the first plate. The strips were developed for 20 minutes at a temperature of 19°C. in the Ferrous Oxalate Developer described by Hurter and Driffield? then fixed, washed, and dried in the usual manner. This developer was chosen because it gives a high density, i.e., develops a maximum number of exposed grains, yet has very little tendency to spread and affect unexposed grains. I t should be pointed out that some later publications in giving the formula for this developer leave out the bromide, which was described by Hurter and Driffield, and the use of which is essential. The densities of the various parts of the plates were measured with a Burt photoelectric cell. This cell was mounted in one end of a dark box, separated by a partition from a tungsten lamp operated at constant voltage in the other end. The plates to be measured were placed over an opening in the partition, the size of which could be adjusted to the size of the section of plate to be measured. A water cell was placed between the lamp and the plate. 4
F. Hurter and V. C. Driffield: Phot. J., 38, 76 (1898).
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RATE OF FORMATION OF THE LATENT IMAGE
The photoelectric cell was connected in series with a 45 volt battery, a condenser, and a charge and discharge key. The key was placed in the charge position for a measured interval of time, then it was thrown into the discharge position and the accumulated charge passed through a ballistic galvanometer of high sensitivity. Comparisons between this photoelectric method and a surface thermopile method gave concordant results for density. The circles on Fig. I represent the developable densities produced by various times of exposure under the conditions described. The abscissa is 2.4
1.8
1.2
0.6
0 0
2 70
90
360
870
26
FIQ.I Time-Density Relations for Thin Emulsions Circles = observed; Solid line = calculated
time of exposure rather than the conventional log time. No correction is made for “fog.” The distance along the time axis for the points t=805; 1305; 1740;2175; and 2 6 1 0 seconds, respectively, are shortened to avoid extension of the graph. The Ideal Maximum Density In the analysis which is to follow we use a concept which we shall call the ideal maximum density. By this we mean the density that the plate would attain if all of the silver ,halide present in the plate had been converted during development into metallic silver. Jones and Hall5 have shown that even a t optimum exposure to light of moderate or low intensity this value is never attained. However, by assuming the relation as demonstrated by Hurter and Driffield6that the density of a plate is proportional to the mass of silver per unit area, it is possible to determine the ideal maximum density by chemical means. In the determination of this value, a plate was cut into four longitudinal strips of precisely equal area. Two alternate strips were exposed to the blue L. A. Jones and V. C. Hall: Proc. Int. Cow. Phot., 7, 1x5 (1928). 6 F.Hurter and V. C. Driffield: Photography, loc. cit. Y
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JULIAN M. BLAIR AND PHILIP A. LEIGHTON
light previously mentioned long enough to produce the maximum developable density for this light. These two strips were then developed with the standard developer, fixed, and washed. Without drying, they were immersed in hot sodium hydroxide solution for 24 hours. The residual silver was then filtered off and washed free from gelatine, dissolved in nitric acid, and the silver precipitated as the chloride and weighed in the usual manner. The two remaining strips of the plate were, without any previous treatment, immersed in hot sodium hydroxide solution for 24 hours. At the end of that time the silver oxide was filtered off and the silver content determined as before. From the relative amounts of silver chloride produced in the two cases the ratio between the ideal maximum density and maximum developable density was determined. The ratio found was 1.095. When applied to the maximum developable density shown in Fig. I , the ideal maximum density for the plates and development conditions used is found to be 2.19.
Derivation of An Equation for Developable Density In attempting an analysis of the rate of formation of the latent image, as indicated by the developable density produced, we have been led to five basic postulates, as follows: I. T h e process ojformatio? ojthe latent image i s chemical. The idea that it probably consists of the photochemical liberation of minute amounts of free metallic silver has received support from the work of many investigators.’ 2. T w o processes are involved, one a forward reaction, tending to produce the latent image, and the other a reverse reaction, tending to destroy it, or to change the developable grains into their original condition. The phenomenon of reversal or solarization is an old and well established fact. The new aspect here presented is that reversal begins as soon as the first grains have been rendered subject to development, and is concurrent with the formation of the latent image. We have obtained direct experimental evidence in support of this assumption, which will be presented in another paper. 3. T h e forward reaction is autocatalytic. This is suggested by the shape of the density-time curve, as well as by the effect of “flash” exposures and by the sensitizing effect of silver on emulsions. 4. The jorward reaction i s a function of the number of grains in the original state. This is a general property of photochemical reactions where light absorption is weak. 5 . T h e reverse reaction i s a junction of the developable density. It is reasonable t o assume that the rate of the reverse reaction, which decreases the number of developable grains, is some function of their concentration, and therefore a function of the developable density. ’Abe9,kAr:hiv Wiias. Phot., I, 268 (189 ), Brit. J. Phot., 46, 196 (1899); LiippoCramer: olloidchemie und Photo ra hie” $908); Lorena and Hiege: 2.anorg. Chem., 92, 27 (1915); Sheppard and Trivefi: !’hot. J., 61, 403 (1921); Fajans: Chem. Ztg., 45, 666 (1921). 9
RATE OF FORMATION OF THE LATENT IMAGE
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Proceeding on the basis of these postulates, we cttn say that the rate of formation of the latent image, dx/dt, will be governed by the relation: dx/dt = fi(X).fZ(b-X) - fs(x)
(1)
where x = the developable density, b = the ideal maximum density, and t = time of exposure to light of constant intensity. Accordingly, fl (x) represents the autocatalytic term, f2(b-x) the term expressing the relation between the forward reaction and the number of grains in the original state, b-x, and fs(x) represents the dependence of the reverse reaction on the number of developable grains. The autocatalytic reaction appears to be proportional, not to x, but to the square root of x, Le., f,(x) = Klx$. The other two functions, since the grains are mutually independent, should involve no more than a direct proportionality. Accordingly, we obtain the equation: dx/dt = Klx$(b-x)
- KZX.
(2)
This equation is found to predict the slope rather accurately for all values of x. I n order to evaluate the constants K1 and Kz, the ratio K1/Kz was found by considering a point on the curve at which maximum density has been at-# tained. At this point dx/dt = 0,x = 2 . 0 and b = 2.19. This gives the ratio
Ki/Kz
= 7.45
By employing this ratio, when other measured values for dx/dt and the corresponding values for x are substituted in the equation, the constant values K1 = 0.0122 and Kz = 0.00164are found. The differential equation (2) may be integrated, and for the determined values of the parameters it yields the relation:
where C is the constant of integration, and possesses the value 2.125. This value of C was determined by substituting corresponding values of x and t for any one point shown in Fig. I. Using this value of C in the above equation, the values of x predicted for various values of t were determined. The resulting curve is shown by the continuous line in Fig. I . Within limits of error, it is in agreement with the experimental curve throughout the whole region of exposures we have thus far investigated. Instead of the usual form of density-time curve, with a considerable “foot” or flat underexposure portion, this curve, both experimental and calculated, has a very short foot, and moreover, strikes the ordinate (zero time) a t a density of C . P .
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JULIAN M. BLAIR AND PHILIP A. LEIGHTON
This is due in part to the fact that no fog correction was made in either the calculation or measurement of x, and in part to the scale of the abscissa, time instead of log time. The thinness of the emulsion must also be a contributory cause of the short foot, since the grains deep in the emulsion receive full exposure and become developable as rapidly as those near the surface. No solarization was observed experimentally and none would be predicted for the values of the parameters found. summary
A differential equation, based on definite theoretical postulates, which expresses the relation between density and time of exposure in photographic emulsions, has been presented. Measurement of the density-exposure relationships on very thin emulsions shows that the equation accurately reproduces the experimental values.
