The Size-Frequency Distribution of Particles of ... - ACS Publications

THE SIZE-FREQUENCY DISTRIBUTION O F PARTICLES O F SILVER HALIDE IN PHOTOGRAPHIC EMULSIONS AND ITS RELATION T O SENSITOMETRIC CHARACTERISTI CS1 PART IV BY

E.

P.

WIGHTMAN, A . P.

H. TRIVELLI,

AND

s. E.

SHEPPARD~

Relation of Size-Frequency to Sensitometric Properties 1. Theoretical Disc.ussion.-Before considering the actual correlation of the observed frequency curves and the sensitometric constants derived from the corresponding characteristic curves, let us see what relations may be anticipated from the photochemical theory of the characteristic curves. These, curves, it is well known, represent the accumulation of the developed silver as a function of exposure, which in turn is a function either of increasing intensity or of increasing time, the other factor being kept constant. If the so-called reciprocity law'

E

= It

held, the two functions would be identical. given by Schwarzschild's law

E'

= It9

(1)

If the exposure is (2)

the intensity and time curves will only differ in the silver mass, or devlsity ordinate by a common factor p , and otherwise will be of the same form. 1 Communication No. 148 from the Research Laboratory of the Eastman Kodak Company. 2 We have given in the preceding paper (Part I11 of this series) the experimental results of size-frequency measurements of a number of typical photographic emulsions and of the determination of their empirical sensitometric characteristics. In the present paper we shall discuss the theoretical bases for the correlation of this data.

142 E. P. Wightman, A. P. H . Trivelli and S. E . Sheppard The conclusions of Slade and Higson that intensity enters into the density-exposure function as I 2 for small values of I (and large values of t ) and as I for large values of I (and small values of t) do not appear to us conclusively proven. In the first place, if uniformity of sensitivity be conditioned by uniformity of size of grain, then their process emulsion does not appear to us, in defect of statistical data, likely to have been sufficiently uniform. Secondly, if sensitivity variation exists among grains of the same size, then their experiments do not really determine “the photochemical law of the (individual) silver halide grain,” for statistical variation of sensitivity among the grains would invalidate any such conception. For the elementary theory it is sufficient, therefore, to assume equation (1) to hold and that either I or t may be held constant while the other is varied. 2. The Fundamental Reaction.-Let us first of all assume the simplest case and set up equations for the course of the fundamental photochemical reaction in a homogeneous system, i. e., molecularly disperse; and then let us consider in what way the derived equations will be modified by the introduction of definite statistical variations associated with the dispersity of the reacting materials. The relations thus obtained will then be typical of the relations to be anticipated between the size-frequency curves and the sensitometric curves. Let A = the thickness of the homogeneous emulsion layer a = the original concentration of the unaffected silver halide x = the amount of silver halide altered in time, t Io = Io’ (1 - R) = the intensity of light incident per unit area, corrected for reflection A = the light absorbed per unit volume K = the absorption c0efficient.l We shall assume initially that the light is monochromatic, that the reaction is irreversible, and also that the product is not photo-active, i. e., neither autocatalytic nor absorptive. 1

R.W. Wood: “Physical Optics.”

Silver Halide in Photographic Emulsions

143

These may well be questioned as undue simplifications, in view of the complexity of the photochemical change in silver halides. Nevertheless, they are the more justifiable, if we consider only the production of the “latent image,” Le., the photo-product which is produced in quantities below visual perception. Yet it has to be assumed that the “latent image’’ production is faithfully revealed by the process of development,’or that D

= y.@ = yx

(3)

.where @ is any given mass of latent image, y is the development factor, i. e., a constant depending only upon the development, and D is the photographic density produced by development. Then according to the Grotthuss-Draper law of photochemical absorption,l the rate of change -dx= -

dt

d(a - x ) = kA dt

(4)

but

where I represents the light transmitted and since from Beers’ law3 1 = Iae-KA(u - .z\ (6) Grotthuss: Gilbert’s Ann. Phys., 61, 50 (1819); Draper: Phil. Mag., cf. also W. D. Bancroft: Jour. Phys. Chem., 12, 213 (1908). If the changed material does absorb then

[a] 19, 195 (1841);

where I has the same value as above and I’ is the intensity of the light absorbed by the changed material with a corresponding absorption coefficient K’. In a paper by F. Halla and A. Schuller [Zeit. phys. Chem., 93, 173 (1918)], they assume a t the start two different absorption coefficients, but they use a somewhat different argument from our own. They later arrived at a result similar to that of Hurter and Driffield (to be considered later) by assuming then that their two absorption coefficients are equal. Beers: ‘Pogg. Ann., 86, 78 (1852). -

144 E. P. Wightwan,A. P. H . Trivelli and S.E . Sheppard equation (5) becomes

hence (4) becomes

Integrating this we have

whence

Let (a

-

x ) = y,

- KA = b, and z

=

and 1-

eby;

then dy =

-b1 . dz z

- i{lns--in(l -z)+c}

eby

when t = 0, then x = 0 and e-KAa

C =-In 1-

e-KAa

substituting this in (10) we get

which, by multiplying numerator and denominator by e-KAx may be put in the form


145

which reduces to e-KAx

= ,-KAR

+ (1 -e--KAR)g--kRId

(13)

Now let us consider the three principal cases (a) when the absorption coefficient, K, is very large, ( b ) the other limiting case when K is very small, and finally, ( c ) the general case for moderate absorption. (a) When K is very large there is very strong absorption. The larger K, the smaller e - K , hence when e-" becomes -z) is also negligibly small, and equation (11) negligible, becomes In eKAx= kKIot (14) which gives 1.

e.,

or, substituting in (3) D

= y.k'Iot

(16)

a linear relation between density and exposure, which is actually true photographically for the first stage of exposure. We should expect this phase to be more prolonged for strongly absorbed (e. g., blue violet) light, and less for slightly absorbed (e. g., red) rays. (b) The alternative extreme case is for very weak absorption, i. e., for K very small-either for very short wave lengths (X-rays) or for very long wave lengths (red, etc., in non-color-sensitized emulsions). Now the function e-KAe can be expanded thus

~

J. Plotnikoff's integration [Zeit. wiss. Phot., 19,238 (1920)lof a similar differential equation leads t o a different result and appears t o us incorrect.

146 E . P. Wightman, A . P. H . Trivelli and S. E . Sheppard

also e-KAr may be expanded in like manner, and if K is very small, we may neglect all terms above the first two, then putting D’,,, for KAa, and D’ for KAx in theseexpansions, and substituting them in equation (12), we get

which may be written

This equation, identical in form with the so-called monomolecular reaction equation for “dark reactions,” is the same as that proposed by Elder1for the characteristic curve, although obtained by slightly different premises, viz., from the so-called “intensity” formulation of reaction velocity of light, and the mass law for a monomolecular reaction.

-dx_- KIo(a - x) dt

It happens that for very low absorption these give the same formula as above. As it stands, however, this doesnot represent the actual course of the sensitometric curve satisfactorily. We shall see later how this equationhas beenmodified to take care of both stronger absorption and statistical variation. (c) Finally, for the general case we may write equation (13) in the form

Hurter and Driffield’s formula2 for the characteristic 1 2

Elder: Jour. Camera Club, 7, 131, 135 (1893). Hurter and Driffield: Jour. SOC.Chem. Ind.,9,455 (1890).


147

curve is obtained from the equation.

'I

@ = 5b - k x - @ - k a i

dt

(23)

Iohaving the same significance as above, i being the inertiaof I the emulsion, ;being approximately equivalent to our

KA; e - k x is the light not absorbed by the changed silver, i. e., the transparency with respect to the changed plate; and e-ka is the light not absorbed by the total silver, i. e., the transparency with respect to the original plate. When integrated and rearranged (23) gives

which is the same as 1

kx = In

[

eka

- (&a

Id

- 1) e - 3 T]

(25)

D '

Hurter and Driffield substituted 7 for kx and 0, i. e., the opacity of the unchanged plate, for eka,thus

Assuming that 0 is large with respect to 1, which is similar to our case (a) where K was assumed large (26) reduces to D = y l n lot y z

(27)

by discarding powers greater than 1. If 0 is assumed very small compared to 1then (26) becomes

It which, unless 7 has a very small value, is very large. But

148 E. P. Wightman, A. P. H . Trivelli and S . E. Sheppard

,

this equation is a linear relation, whereas in case ( b ) above we obtain the monomolecular relation. It does not appear worth while a t present to discuss in detail the differences or similarities of these equations. Ross1has pointed out, however, that Hurter and Driffield’s assumption that the transparency of the light-affected material is given by e-kr, while that of unaffectedmaterialisgiven by e-ka, must be incorrect, since it leads to a mathematical inconsistency. The incorrectness appears to lie in the assumption that k , the absorption coefficient, is the same for both affected and unaffected halide. The experimental proof Hurter and Driffield give for this is vitiated by the difficulty of determining true absorption coefficients of silver halide from opacities of emulsions. On the other hand, the assumptions*we have made in the general formula, namely, that the photo-product exerts neither a light absorbing nor a catalytic rdle, are not entirely warranted. However, the complications introduced by allowing for these are for the moment not justified by any obvious advantages. 3. Eject of Heterogeneity of Material on the Fundamental Reaction.-Before considering any such complications of the fundamental reaction equation, let us rather suppose this to be of simple type, e. g., either like equation (16) or (19), and then consider the effect produced by supposing the material not homogeneous and continuous, but heterogeneous and discrete. In point of fact, the silver halide emulsion has not the homogeneity really assumed in the Hurter and Driffield and Ross: Jour. Opt. SOC.Amer., 4,255 (1920). If the actual photosensitive material be supposed to be dispersed metallic silver, in agreement (to this extent a t least) with Renwick’s theory of the latent image [Hurter Memorial Lecture, Jour. SOC. Chem. Ind., 39, 156%’ (1920)] and with Weigert’s recent work [Sitzungsber. Akad. Wiss. Berlin, 39,641 (1921)], and if we suppose this adds to itself autocatalytically, by action on the silver halide, then the autocatalysis will tend t o be opposed by the increased absorption, thus approaching the condition assumed above (so long as we are considering only the latent image). 1


149

other theories, which suppose a molecularly disperse system, each molecule of identical reactivity to light. Actually, the material is a suspension of crystalline grains of a variety of sizes-if not varying in other respects. It may be further considered possible, even probable, that wide variations of “sensitivity” exist among the grains of one and the same emulsion. We may initially define “sensitivity” as the reciprocal of the energy required to make a grain developable by a developer of given reduction potential. It is generally accepted at present that the condition for developability is a nucleus, i. e., that the photo-effect, whatever its nature, is localized in the grain. The most concrete development of this is that the nuclei consist of colloid silver, either provided (in part a t least) by photolysis of silver halide, or formed by condensation of pre-existent colloid silver amicrons to submicrons. This latter hypothesis is due to F. F. Renwick. His primary contention, that the sensitive substance is not silver halide a t all, but colloid (metallic) silver has recently received support from some remarkable experiments of F. Weigert.2 In formal contrast to this stands the thesis that the photo-effect is not localized in nuclei, but consists in the altered reactivity of the silver halide crystal as a whole, or at least of a whole surface. This increased reactivity may now be conceived as a change of quantum state of the atoms, e. g., in such a form that the ionic pairs Ag+Brin the lattice system [Ag Brs] [Ag, Br] possess intermediate stable states intervening in the reaction

Br-

0

I

Ag+ I

Br (bromine atom)

Ag (silver atom)

J.

_ic

J.

and that these intermediate phases have increased reactivity (e. g., by way of adsorption) toward reducing agents. The F. F. Renwick: Hurter Memorial Lecture, Jour. SOC. Chem. Ind., 39, 156T (1920). 2 F. Weigert: Sitzungsber. Akad. Wiss. Berlin, 39, 641 (1921).

150 E . P. W i g h t m a n , A. P. H . Trivelli and S. E. Sheppard behavior of the latent image to oxidizing agents, and its stability when silver bromide is converted to iodide are not satisfactorily accounted for. In either case, if we attribute a definite sensitivity to a grain, this means that a certain finite increment of energy is necessary for developability . In the general problem of chemical reactivity, what has been termed the critical energy increment1 of a molecule is analogous to this “inertia” of the grain, the reciprocal of its sensitivity. Considering the variable of sensitometry, the density D, it must be remembered that to a near approximation grains contribute, on complete development, to density in proportion to their number and size conjointly, or again, to a slightly lower approximation, to their number and projective area. The quotient AD

density increment

aE = energy increment

reckoned per unit area, would appear a p i o r i to be independent of the size of grain, assuming a continuous wave-front. But if conditions permit a local concentration of energy related to the size of grain, the magnitude of the above quotient will depend upon the distribution of grain sizes in unit area. Supposing a continuous wave-front, that is, not a quantumlike or discrete structure of radiation, local condensation of the energy to the critical increment might be effected in the material: (1) by reason of a variable distribution of a photochemical catalyst in the material, or specifically, among the grains. The probability of a grain reaching the critical increment would then depend upon its having a certain amount, and perhaps individual distribution, of the catalyst, the primary function of which would be to increase the energy density of absorbable type.z Colloid silver submicrons supply a concrete example. Alternatively, it may be supposed that the silver halide is itself relatively inert, and with Renwick that the colloid silver particles are the true photosensitive material. In either case, 1 2

W. Mc C. Lewis: “System of Physical Chemistry,” 3, 140 (1919). W. Mc C. Lewis: LOC.cit.


151

the distribution of the “catalyst” could depend on the size of the grains, e. g., the probability of a grain having sufficient catalyst might be proportional to its mass. I n this case, the sensitiveness-variation curve would coincide, to a first approximation, with the size-frequency curve, for one and the same emulsion. On the other hand, the effective distribution of catalyst might not coincide with the size-frequency. The very ingenious experiment of Toy1 shows that grains of the same size and shape similarly oriented to the light need not have the same sensitiveness but can give a normal characteristic curve. (2) Local condensation and a sensitivity variation might be effected, as suggested in Toy’s paper, by optical conditions of refraction, reflection, etc., depending on chance orientations and concentrations of the crystalline grains. Such a hypothesis is difficult to prove or disprove, but the statistical data being accumulated may strengthen or weaken its plausibility. There remains, by exhaustion, the possibility of local, or granular, concentration of energy not by virtue of inherent factors of the sensitive material, nor by accidental circumstances in its disposition, but as a necessary consequence of a discrete structure of the energy itselfq2 The mathematical development of this assumption has been It may be pointed out here dealt with by Dr. L. Silber~tein.~ that to a large extent it gives the same relation between effective sensitivity variation and size-frequency distribution as F. C. Toy: Phot. Jour., 61, 417 (1921). That such grains can give a characteristic curve indicates either that they vary in sensitiveness inherently from one another by other factors than size and shape, or that the fundamenta1 reaction is governed by none of the premises so far assumed-namely, continuous wave front, etc., and would apparently favor the quantum hypothesis mentioned in the next succeeding paragraph. On the other hand, the fact that there are emulsions of large grain size which are less sensitive than some emulsions with smaller sizes, and also that an emulsion can change its sensitiveness on standing unexposed for a period of time in its original box seem to show t h a t we cannot throw overboard every idea of a n inherent variation in sensitiveness. We may note here also, in connection with the data given later, that in thick grains the position of a photocatalytic center might vary in grains of the same size and shape. * C f . F. E. Ross: Astrophys. Jour., 52, 96 (1920). L. Siberstein: Phil. Mag. in press (1922).

152 E. P. Wightman,A. P. H . Trivelli and S. E. Sheppard that previously discussed on the hypothesis of a photocatalyst distributed among grains with a probability varying as their masses. On the quantum theory, however, it is the projective area which is fundamental rather than the mass. The very important modifying factor of “clumping,” or secondary aggregation of the primary grains will be noted in connection with the results of measurement of size-frequency. Instead of making any restricted hypothesis concerning sensitivity, we may suppose that a general sensitivity variation exists, without explicit statement as to what determines sensitivity. The conception of statistical variation among the reacting units-as distinct from difference of chemical specieshas not entered much into the general chemical dynamics. Statistical variation of gas molecules as to velocity has been from the first, however, a feature of the kinetic theory of gases. Through statistical mechanics and the radiation theory this has begun to react powerfully on classical chemical kinetics. . Nevertheless, it is not in this field that we may immediately expect to find instances of variation applied to reaction velocity problems, but rather in the field of biochemistry. In problems of the immunochemistry of cells and unicellular organisms, the existence of variation has to be admitted ex post facto, without, at present’, any satisfactory theory for the origin of variation prevailing. While we have utilized conceptions developed in this field by W. McC. Lewis, R. C. Tolman and E. K. Rideal, without recitation of the earlier work, we may refer particularly to a valuable review by S. C. Brooks,l on “The Mechanism of Disinfection and Haemolysis.” In order not to confuse the issue, we shall not refer further to the explicit cases dealt with by this author (haemolysisof blood corpuscles, mortality of spores, etc., under attack of disinfectants), but briefly restate his argument for the general case of a reaction-termed the fundamental reaction-consisting in the transformation of one kind of discrete particles into another-in oyr own case, the passage of ‘Brooks: Jour. Gen. Physiol., 1, 61(1919).


153

unexposed silver halide grains into developable grains, and thereby, into silver grains, under the influence of light. So far we have discussed such a change solely in terms of the course of a fundamental reaction-implicitly of identical molecules not varying inter se. Brooks points out that the actual course of such a process may be very largely dependent upon the variations in resistance among the different individuals, and only secondarily upon the course of the fundamental reaction. Now the actual progress of the reaction may be expressed graphically in two ways. First, as the usual accumulation or time-curve, the amount changed being plotted as ordinate against time as abscissa. For dispersed granules or cells, we could plot (number x volume) against time. There is also the rate curve, the ordinates being the amount of substance Ax At

changed per unit time -, plotted against time. The former is the more usual in reaction kinetics, while the latter is the “mortality curve” of vital statistics. The latter curve is evidently the first derivative of the time curve, and we can pass from the rate curve to the tiwe curve, either empirically by graphic integration, or algebraically, if the differential equation expressing the rate curve is known and integrable. Conversely we can obtain the rate curve from the time curve by differentiation. Essential to the present argument is the fact that under certain conditions the rate (or mortality) curve may be identical with the variation (or frequency) curve of individual resistivities (inertias, reciprocals of sensitivities). Supposing the time-curve replaced photographically by an exposure curve

then with certain restrictions to be discussed, the first derivative of this dD

=

f’ (E)

154 E. P. Wightman,A.P. H . Trivelli and S. E. Sheppard would give the variation curve of inertias of the reacting particles, or correlatively, the sensitivity-frequency curve. The importance of the first derivative of the characteristic curve was first noticed by Nutting, Jones and Meesl for positive emulsions. It was introduced as dependent upon finite groups of grain sensitivities by Ross2 in a theoretical discussion of the density-exposure function, and attention was a little later drawn to the subject in respect to negative emulsion ~ latter’s statement of the fundaby F. F. R e n ~ i c k . The mental property of the derived curve appears to agree to a large extent with the conclusions reached here. “Since every ordinate of the derived curve represents the amount of silver bromide which has just attained the developable condition a t the corresponding exposure, it is evident that we might regard each ordinate as a measure of the quantity of silver bromide in the film having that particular true inertia value.” It is now evident that if for “quantity” we substitute “number of particles per unit area” that the first derivative becomes the statistical frequency curve of inertias-provided there is no reduction of intensity through the film, as noted by Renwick, but also provided that the relation of grain size to grain sensitiveness is explicit. Considerable modifications are necessary before these premises can be applied to actual sensitometric curves-obtained normally as D = 4 (log E)

(31)

with first derivative (32)

To facilitate this application we have developed an approximate theory of the density-exposure function for very restricted conditions. The direct theoretical development of a 2

Phot. Jour., 54, 342 (1914). F. E. Ross: Jour. Opt. SOC.Amer., 4, 255 (1920). Phot. Jour., 61, 10 (1921).


155

function for the characteristic curve is beset with difficulty. Application of the calculus to a system composed of finite discontinuous units, such as silver halide grains, is itself contestable, and the difficulty becomes greater if statistical variation of the grain sensitivities is admitted. The following device has been adopted for an indirect attack. In the relatively simple photochemical reaction in an ideal homogeneous system, as assumed a t the outset, the heterogeneity or variation factor will now be introduced by replacing the reaction-constant of this function by various sensitivity or inertia-variations. These expressions, on the other hand, are such as have been found experimentally to represent our size-frequency data. A preliminary basis is afforded for comparison of these sizefrequency curves en bloc :a. With graphically determined first derivatives of experimental D = f (E) curves.

b. With algebraically derived first derivatives of theoretical D =f

(E) curves.

We do not propose a t this stage to consider the effect of modifying assumptions proper to the homogeneous reaction, e. g., allowing for absorption by the altered silver halide, for limited autocatalysis by the photoproduct, etc.l It is indeed quite feasible that actually, for any grain singly, the absorption by altered halide plays a negligible part for the normal latent image formation, since the “latent image” active in normal development will be mainly superficial. It has appeared to us more important to proceed to consideration of sensitivity variation. Consider the simplest case of sensitivity distribution, viz., negligible deviation from a certain average value. The The changes in developability of the grain, with different developers, with solarization, with iodide, etc., are not consistent with so simple a view as that adopted, but these are secondary properties.

156 E . P. Wightman,A.P. H . Trivelli and S. E . Sheppard

sensitivity variation curve, corresponding to the rate curve, would be a straight line perpendicular to the abscissa. Its integral, the time curve, would, if the fundamental reaction is supposed uniform (of zero order), follow the axis to this point of intersection, then rise perpendicularly. That is, all grains would become developable a t the same exposure. Apart from inherent and “quantum” sensitivity variations, in thick layers absorption of light in the upper layers involves an apparent gradient of sensitivity through the film. This is partly allowed for in the deduction of a reaction formula, which remains, however, of complicated form except for very high or very low values of the absorption factor. (See previous section.) Following Ross, we may combine this absorption factor with inherent variation of sensitivity. Before resorting to this artifice it will be simpler to limit consideration a t first to onegrain layers. Suppose the photochemical absorption proper to be of such large or such small magnitude as to give the limiting cases obtained for the ideal reaction : (a)

(b)

x =

kE,Uniform rate

x = a (1 - e-k*), Monomolecular rate.

(33)

(34)

Then let us assume that the material is broken up into “microsystems,” each obeying one or the other of above formulae but in either case having different individual values of kthe velocity constant-distributed according to some probability function, e. g., the normal law of error. We must then equate the actual rate to the distribution function of the k f values, i. e,, to the sensitivity distributi0n.l For immediate convenience in testing the “size variation” hypothesis of sensitivity, we have selected distribution functions of the type found experimentally to represent our actual size-frequency data. The general progressive increase of “speed” with grain 1 The equating of i l/k to E = .+ (I, t ) will appear more obvious if we reflect that i ( = resistance or inertia) is effectively the period of existence of t h e particle unchanged by light, i. e., is the exposure required to change it.

.


157

size shown in our data (Jour. Phys. Chern., 27, 1 (1923). warrants such an attempt. In general, if the granular material does not cover the whole of a given area, say 1 sq. cm., for which the reaction is being considered, the available exposure will be that falling on the normal grain surface or projective area, i. e., according to previous considerations.

2

_ -- k1.S

a.

edt

b.

= k’I.S(a

- x); x

x

=

S.E

= a(I-e-k’SE)

(35)

(36)

provided the grains have the same sensitivities (or inertias), n

and S =

s, the sum of the areas of n grains per sq. cm. Now 0

we may obviously have two extreme cases for the function

k

=

f ( i ) , the variation of inertia, in relation to size of

grain. These are: Case (i) k = f(i),totally independent of grain-size. Then, if the grains are the Same size, integration has to be extended only to inertias, if of different sizes, to both inertias and grain sizes. Case (ii) k = f ( i ) is a function of grain size. For example, the sensitivity may vary directly as the grain-size, or the inertia inversely proportional thereto. This is the simplest possibility, integration over a range of sizes being at the same time integration over sensitivities. According to the argument developed, either of these cases may be conjoined with the fundamental “molecular” equation ( a ) or ( b ). This procedure is developed to illustrate the modifications, in the following synopsis:

158 E . P. Wightman,A . P. H . Trivelliand S. E. Sheppard

. EQUATION TYPE(a) Fundamental reaction

ddxE = k

Variation function of inertias

k=y=yo

...

Integral, or exposure function

s

(A11 same size)

i. dD a = k

Size condition

f(i)= y = yo

S

D = LEBs

k

S

D =s b d i

zz

dx=k dE

= f(i)

e. g.,f(i) =Be-k’

S

Note.-A uniform rate of reaction, combined with a n exponential inertiadistribution. gives the same type of density function (for constant grain size), as the “mono-molecular” rate combined with uniform inertia-distribution. If the size of grain s varies, integration must be extended over f (s) ds for the corresponding limits as f (i) &-which are not given unless the relation of i t o s is explicit.


159

EQUATION TYPE(b) Fundamental reaction

i** k(a--i) dE =

Variation funcSize tion of inertias condition

S

k constant

Integral, or exposure function

D

=

Dm(l-e-k'E)

(Same size)

S

or

Dm

D, k' ('-8

- k'E)

(D, - D) = e

Note.-This

expression, D = D,

D=D,

[l-e-="

(1-e-")

which may be written

p E d E ]

has considerable similarity to that obtained by Ross' ('Jour. Opt. Soc. Amer., 4, 255 (1920)), viz.

where n is the finite number of classes in grains of different sensitivities supposed arranged in geometrical progression, p the sensitivity factor, r the ratio of sensi,tivities of different groups.

160 E. P. Wightman,A . P. H . Trivelli and S. E . Sheppard Relation of Size of Grain to Inertia On the “quantum” theory, as developed by Dr. I,. Silberstein, the relation of “size of grain’’ to “inertia of grain” is apparently quite explicit. The projective-area of the grain, say s, is the measure of its chance of being hit by a quantum, hence, is the effective sensitiveness, statistically. On the other hand, the “size of grain” considered in this is not necessarily the “size” as found by the direct measurements of frequency, nor is the distribution function of sizes thus directly available for the quantum theory but requires some corrections for reasons noted later. We may, without immediate recourse to this theory, test empirically the hypothesis that the effective sensitivity of a grain is its “size”-to a first approximation, its “area.” Then, the value of f ( i ) in the foregoing synopses will be given by the distribution of reciprocals of the areas of the grains. On this assumption, the relation of size of grain to inertia is explicit; if the reciprocal area-frequency ( = inertia-frequency) per sq. cm. of plate be multiplied at each point by the area itself, the resultant curve should give, to a first approximation, the first derivation of the density-exposure function, when expressed in suitable units. Such curves are given in Fig. 1 as constructed from the original “frequency curve” for the two types of emulsion in question. They are each plotted for two cases, A. for the original multilayer plate. B. for a one-grain layer plate. If now the curves (Fig. 1,No. 1,A and B and No. 2, A and B) are integrated graphically, the resulting curves should be comparable with the D-E curves. This has been done for both concentrated and dilute experimental emulsions and the results are shown in Figs. 2 and 3. (See Table I for data.) It is obvious that for adequate comparison, both abscissae and ordinates should be expressed in the same units in both cases. The scale for the exposures, is obtained from the condition that the maximum density must be reached with exhaustion of all classes of grains. For the density ordinates, the values 2 and 2’ in Figs. 2 and 3 should be multiplied by a factor for con-


I

Fig. 1

EXPER. EMULS. NO.l COMC.

Fig. 2

EXPER. no. I

161

162 E . P . Wightman,A.P. H . Trivelli and S. E . Sheppard

n

h X

EXPER. EMULS. DIL.

- -

1 k 1'- DENS. EXP. 2 &2 ' - INT. AREA REG AREA

Fig 3


163

version of silver halide of given grain size to density, a t same gamma as in Fig. 4. This correction will be applied when present work on the photometric constant is completed. It will be seen that the “synthetic”curves give a “cut,” as found by Renwick, but a very prolonged underexposure region, and do not show very satisfactory correspondence with the actual sensitometric curves. The very prolonged “underexposure” effect corresponds, however, to the fact that no allowance has been made for reduction of available light energy in the thickness of the film, scattering of light and other sects. With these corrections it appears possible that the comparison may be improved. Ideally, it would seem that the one-grain layer would be the most satisfactory for comparison. The difficulty lies in the measurement of very small “densities”-of the order 0.002 to 0.20-with sufficient precision. Work on this is in prog-

TABLE I Experimental Emulsions Nos. 1 and 2

-1 X

=qli

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0

No. 2

No. 1

:nteg. area in p: [nteg. area in pz [nteg. area in pz hteg. area in 12 X 10- 7 (conc.) X10-6 (dil.) X 10- 7 (conc.) X 10- 6 (dil.)

-

1.5 3.0 5.5 10.5 15.5 20.5 26.5 32.5 38.5 44.5 51 58 64 70 77 88 89

0.53 1.06 1.9 3.7 5.5 7.2 9.3 11.5 13.6 15.6 17.9 20.2 22.3 24.6 27.1 29.3 31.4

0.3 6.9 20.4 34.5 46.8 56.1 64.5 72.6 79.8 85.5 90.3 94.8 99.6 104.7 108.0 110.7 114.3 116.7

0.75 17.3 51.0 86.3 117 140 161 182 200 214 226 237 249 262 270 277 281 292

164 E. P. Wightman,A . P. H . Trivelli and S . E. Sheppard

TABLE I (Continued) Experimental Emulsions Nos. 1 and 2

I E 1.5

11-

i.6 i.7 i.8 i.9 i.95 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

;::1 2.2 2.3 2.4 2.5 2.6

2.7 2.8

I

E

0.891 1.00 1.26 1.59 2.00 2.51 3.16 3.98 5.01 6.31 7.94 10.00 12.59 15.9 20.0 25.1 31.6 39.8 50.1 63.1 79.4 100

200 251 316 398

I

No. 1

No. 2

Dkonc.) D(di1.) D(conc.) D (dil.) I(frorn graph)/(frorn graph)/(fromgraph)l(frgm graph)

0.002 0.01 0.02 0.06 0.105 0.18 0.25 0.34 0.46 0.60 0.78 0.95 1.12 1.30 1.48

1.65 1.80 1.92 2.01

-

-

0.03 0.033 0.036 0.04 0.043 0.046 0.05 0.056 0.063 0.07 0.078 0.086 0.095 0.104 O.lt4 0.125 0.136 0.148 0.160 ,O .173 0.186 0.200 0.213 0.226 0.240 0.253 I 0.266 0.280

I

0.180 0.180 0.185 0.201 0.232 0,28 0.34 0.41 0.49 0.60 0.72 0.86 1.01 1.16 1.31 1.46 1.60 1.72 1.83 1.93 2.02 2.09

-

-

l-

-

--

,-

-

0.02 0.021 0.022 0.024 0.026 0.029 0.032 0.036 0.040 0.045 0.053 0.062 0.071 0.079 0.088 0.097 0.105 0.114 0.123 0.131 0.140 0.149 0.157 0.166 0.175 ~-

.

-

ress, and a more satisfactory comparison will then be possible. These relations between the size-frequency curves and the density-exposure function are admittedly tentative. A sta-


165

tistical relation between size of grain and sensitiveness may exist, as already noted, either in consequence of a corresponding distribution of photocatalyst, e. g., colloid silver, or as a result of a discrete structure of radiation. The possibility of the sensitiveness of the grains varying with their sizes in consequence of content of colloid silver may be illustrated as follows. It is well known that the spectral absorption of colloid silver sols varies with the (average) size of particle. Suppose a series of absorption curves plotted for increasing average size of colloid particles. At any given wave-length, for each grade of colloid silver, a different value of the specific absorption coefficient will exist. It is feasible that in crystals of different sizes in one emulsion adsorbed colloid silver might be present in different grades, e. g., it might be more condensed the greater the size of the crysta1.l Consequently, we should have a value of the absorption coefficient varying with the size of grain. Alternatively, only the mass and not the distribution of colloidal silver might vary. These possibilities are given by way of illustration; meanwhile, experimental tests will be made by following microscopically the action of desensitizing (oxidizing) agents on one-grain layers. I t appears possible that this may yield a crucial test as between the “photocatalytic” and the “quantum” hypotheses. A statistical distribution of “photocatalyst” could give results to a certain point, similar to a quantum theory, but desensitizing (by destruction of photocatalyst) should differentiate between them.

Summary 1. The existence and nature of statistical variation of sensitivity among silver halide grains is discussed, 2. A mathematical expression for photolysis in a homogeneous system is developed in relation to the density-exposure function. Compare certain considerations advanced in a recent paper on “The Action of Soluble Iodides and Cyanides on the Photographic Emulsion” by S. E. Sheppard: Phot. Jour., 62, 89 (1922) and in “The Silver Halide Grain of Photographic Emulsions” by A. P. H. Trivelli and S. E. Slieppard: ibid., pp. 55-56.

166 E . P . Wightman, A . P . H . Trivslli and S. E. Skeppard 3. The relation of statistical variation of inertia to the density-exposure function is discussed. It is concluded that under certain conditions the first derivative of the density, exposure function will correspond with the inertia-variation function or curve. 4. Conditions permitting this correspondence are noted ; if the size-frequency correspond with the sensitivity-fr equency, then the curve of “reciprocal size-frequency” gives the inertiavariation curve, integration of this the density-exposure function. Rochester, N . Y May I , I922

The Size-Frequency Distribution of Particles of ... - ACS Publications

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