Developer Temperature Compensation Formula

Photo Engineer · Jul 20, 2008

Patrick;

I followed you completely. What you appear to have missed in my post is that you can only do this for one film and one developer due to the variations in thickness (difffusion related), silver coverage, and emulsion type and for the developer you have to consider developing agent(s), concentrations, ratios, pH and other addenda.

Simple substitution of Benzotriazole for bromide as antifoggant can upset this entire scheme of things. I have plots that show some developers to be nearly parallel and others to be big outliers. I think there is an earlier set of curves posted here that you may wish to look at.

So you are right for one film and one developer but you cannot use that same equation for multiple films and developers, only the methodology holds true. So, I would use your method for time/temp for PX, SXX, TX and etc in one developer (which might work) or PX in multiple developers (again which might work), but I wouldn't cross the data over. Kodak has shown data to support my position in the book I noted earler.

PE

df cardwell · Jul 20, 2008

so curves for different temperatures emanate from the same point and are also straight lines.

Gadget,
Does this chart agree with your graphing ? I'm not quite following what you're saying, and am a little confused. Well, confused.

don

Photo Engineer · Jul 20, 2008

Don;

He is using semi log paper to plot the same generic type of data, but with a twist as he is varying contrast. The Kodak and your data and mine are based on "aim" contrast. Our data also compare a film in a developer or a group of developers. Or a group of films with a developer to show the curves you have.

If you look at your 3 films, they just don't match each other! That is the key to this. Each film and each developer is different. Patrick is just doing something different in that his example, as I said, has varying contrasts.

PE

df cardwell · Jul 20, 2008

Thanks, PE & Patrick

I'll go back to the darkroom now !

don

gainer · Jul 21, 2008

Photo Engineer said:
Patrick;

I followed you completely. What you appear to have missed in my post is that you can only do this for one film and one developer due to the variations in thickness (difffusion related), silver coverage, and emulsion type and for the developer you have to consider developing agent(s), concentrations, ratios, pH and other addenda.

Simple substitution of Benzotriazole for bromide as antifoggant can upset this entire scheme of things. I have plots that show some developers to be nearly parallel and others to be big outliers. I think there is an earlier set of curves posted here that you may wish to look at.

So you are right for one film and one developer but you cannot use that same equation for multiple films and developers, only the methodology holds true. So, I would use your method for time/temp for PX, SXX, TX and etc in one developer (which might work) or PX in multiple developers (again which might work), but I wouldn't cross the data over. Kodak has shown data to support my position in the book I noted earler.

PE

I don't know what you mean by "the same equation." If you mean the constants in the equation vary with each film-developer combination, you are right. I never said otherwise. If you mean that the form of the equation is different for each case, I have proved otherwise by using that form to fit every film that Kodak reported in the time-temperature tables that Kodak published soon after XTOL hit the market, which included the Ilford films and others as well as the then-current TMAX films. The same general equation fits an exponential decay function. I guess I'll have to find a way to post the whole article.

Photo Engineer · Jul 21, 2008

Patrick;

If you look at some of the premises here, they present a single equation to do just that, and that is the point of the whole argument. I am agreeing with you and disagrring with them. There is even an excel spreadsheet posted here that purports to do just what you and I say is impossible.

You may have missed those posts.

PE

gainer · Jul 21, 2008

I did not come up with the equations I used without some basis. According to Hardy & Perrin in "Principles of Optics":
G = Gmax - Gmax * e^(-kt)]
or,
Gmax - G = Gmax * e^(-kt)
where G is gamma and Gmax is gamma infinity. Taking the logarithm of both sides,
log(Gmax - G) = log(G max) - kt
Thus, a plot of log(Gmax -G) vs time of development should be a straight line with negative slope k. Likewise, a plot of Gmax-G vs time on semilog graph paper should be a straight line. The rub is that we must know the value of Gmax. If the results of two development tests at time such that one is twice the other, then:
Gmax = (G1^2)/(2*G1 - G2).

(For the uninitiated, read that "Gmax = G1 squared divided by the quantity (2 times G1 - G2).)

If any of our current films were not to follow this form, I would think it would be the Tmax type. They were among the closest to this form.

I did not say or show in any way that development time for constant G is a straight line. What I showed was that over the range of data available, the gradient is a linear function of temperature on semilog paper when development time is held constant. Look closely at the chart I provided. There is no implication or statement that I consider the same numerical relationship to be valid for any other combination of film and developer. The linear relationship between temperature and log(G) at constant time has held for all the films and developers I have tested, including those that Phil Davis printed in Photo Techniques (probably Darkroom Techniques then). If you read the times at constant log(G) from my chart, you will see that there is definitely NOT a linear relationship. What are we really arguing about?

gainer · Jul 21, 2008

df cardwell said:
so curves for different temperatures emanate from the same point and are also straight lines.

Gadget,
Does this chart agree with your graphing ? I'm not quite following what you're saying, and am a little confused. Well, confused.

don

The premise in the Rodinal curves is that you are trying to hold a constant gamma by adjusting time to suit temperature. My curves show the effects of time and temperature on gamma, but the "mapping" is not conventional. Each line on my plot is for a constant temperature. Each line on the Rodinal chart is for a constant gamma. I will look in more detail to see what kind of a chart I can get from the one you posted. I used to have a copy of that chart.

gainer · Jul 21, 2008

I used the Agfa data for 20 C, 12 m, G=0.55, 16.8 m, G=0.65, to estimate that the Gmax for APX100 is 0.834. I had to use my program for successive approximation to get it. This establishes a line on semilog paper. I used a point at 22 C, G=0.65, t=14 m to establish another line. Now I have two lines on semilog paper emanating from 0.834 at t=0, one passing through (0.834 - 0.65) at 16.8 m and the other through (0.834 - 0.65) at 14 m. G is on the log axis and t is on the linear axis.

In order to establish lines for 18 C and 24 C, I drew a vertical line across the other lines. I marked a point along this line above the 20 degree line a distance equal to the distance along the vertical line between the 20 and 22 degree lines, and another point the same distance below the 20 degree line. I now have 4 constant temperature lines emanating from the same Gmax. All of these were constructed from the 2 pairs of time-gamma values. That of course does not make them valid representations of nature, but there are data on the Rodinal chart I have not used yet. The value of G at 20 C, 18.8 minutes should be 0.75, according to Agfa's chart. I see a value of Gmax-G of about 0.1. 0.834 -.1 = 0.734 in the chart I made.

gainer · Jul 23, 2008

It occurred to me that some may have gotten the idea that I was trying to generalize this analysis too much. I would not have done it if I had not in fact realized that all films are different in particulars. At the same time, there are similarities that have been recognized since before the days of hurter & Driffield, Hardy & Perrin, and many other well known and respected researchers into the mysteries of photography. My premise was that I should be able to use the similarities to minimize the amount of testing needed to define the specific differences among films.

df cardwell · Jul 23, 2008

Gadget

You make my brain hurt, but I kinda like it. I'll let it simmer, and then let it perc.

Thanks

don

Photo Engineer · Jul 23, 2008

Imagine a thin dupe or print film using AgCl/Br emulsions with a 0.2 micron grain and compare it to a thick 10 micron, AgBr/I emulsion with 10% iodide. The two emulsions will have the following properties in common at the start: Native blue sensitivity and a curve that follows a cubic spline in shape called an H&D or characteristic curve.

They will differ in granularity, sharpness, distribution of sensitivity, development rate vs developer, temperature sensitivity and edge effects (sharpness) for starters. As noted in previously posted curves, the development rate vs temperature follows a semi log scale to yield a straight line, but each of these will be different.

DF, you have it all under your hat. Nothing to worry about!

PE

df cardwell · Jul 23, 2008

I tripped over the cubic spline,
but I think I'll be better after a little beer.

Thanks PE

Photo Engineer · Jul 23, 2008

The mathematical equation for a common characteristic curve (H&D curve) is called a cubic spline.

PE

df cardwell · Jul 23, 2008

Photo Engineer said:
The mathematical equation for a common characteristic curve (H&D curve) is called a cubic spline.

PE

BACK TO THE BOOKS !

Photo Engineer · Jul 23, 2008

DF;

Sorry to do this to you but: http://en.wikipedia.org/wiki/Spline_(mathematics) and that is why they call us photo engineers.

This URL will only work from Google for some reason. It goes to a generic math page otherwise. I have not been able to repair the URL. Sorry.

Found the problem. For some reason, the final parenthesis is missing in the redirection. Just add the parenthesis and it will work. APUG is parsing it wrong.

PE

srs5694 · Jul 23, 2008

Photo Engineer said:
DF;

Sorry to do this to you but: http://en.wikipedia.org/wiki/Spline_(mathematics) and that is why they call us photo engineers.

This URL will only work from Google for some reason. It goes to a generic math page otherwise. I have not been able to repair the URL. Sorry.

Found the problem. For some reason, the final parenthesis is missing in the redirection. Just add the parenthesis and it will work. APUG is parsing it wrong.

In replying with a quote, I see the closing parenthesis after the end-of-URL tag. Let's see if that's the software re-writing the code by editing and re-posting:

http://en.wikipedia.org/wiki/Spline_(mathematics)

Yup, looks OK to me. Sorry to say it looks like human error to begin with....

gainer · Jul 23, 2008

How did we get into trying to fit a curve to the film's characteristic curve? I never tried to do that. Neither did Kodak when it published j109 (IIRC) which presented development charts for just about every film available then for XTOL. Those charts presented CI values, which are slopes of straight lines, the definition of which I used to know when we were under ASA. They still are, I think, defined pretty much the same way, including a specified part of the toe. To put a little different slant on what I have, my complete program will take two of the numbers on one of those charts at one temperature and one number at a different temperature and recreate the whole chart. That means, of course, that if I develop 3 grey scale negatives, 2 at one temperature but different times and the third at any time but a different temperature, I can make my own chart that shows times for 5 or more values of CI at 5 or more different temperatures that will be just as useful as the ones Kodak presented, but for any combination of film and developer I choose. The actual shape of the characteristic curve only enters in when deciding where to start the straight line that allows a single number for CI. Kodak does that for me. If I want to do the same kind of chart for a developer that is not XTOL, I can use my own definition of CI without messing with arbitrary curve shapes. The cubic spline is usable for approximating many diferent functions, but an experienced photographer will know where he wants to start the toe of his characteristic curve. It will probably be 0.1 above base + fog. Fooey on splines.

alanrockwood · Jul 24, 2008

I highly doubt that a cubic spline function is fundamental to an exposure and density curve. In fact, I could almost guarantee that it is not. For one thing, a cubic spline is a piece-wise continuous function, with a few other nice mathematical properties thrown in, like slope matching at the segment boundaries. Very very few physical processes follow a piece-wise continuous functional form, let alone one based on piece-wise cubic polynomials. For another, a cubic function has the wrong behavior in the asymptotic limit of low exposure, where the curve levels off and asymptotically approaches a level curve. Polynomials do not have this property, regardless of whether they are used in piece-wise fashion or not.

What a cubic spline is good for is to empirically fit a smooth function to experimental data or to another function, either by interpolation (the curve going through every point) or by a curve fitting approximation (the curve passing in the neighborhood of as many points as possible, though not necessarily through them.) A cubic spline works pretty well for this (amazingly well in many cases), and would be very practical for fitting exposure density curves. However, the good fit is not because cubic splines fundamentally describe the underlying processes, but only because cubic splines are very flexible and good at approximating almost any reasonably smooth function, provided one picks an appropriate number of segments for the cubic spline curve.

Sorry to get too fancy in the description, but I think it is worth pointing out some of the properties of cubic splines in order to clear up possible misunderstandings.

Photo Engineer · Jul 24, 2008

Well, here are two answers:

Patrick; I am not relating what we have been discussing to the curve of film, but rather to illustrate how different each film behaves regarding its kinetics and how each developer also behaves according to its kinetics. To expect the kinetics of a pair of films and developers picked at random to behave according to the same kinetics and the same thermodynamics is not possible. The curve of the film is just one characteristic used to prove that it is impossible because the curves obtained will not mach.

Alan;

I agree, but since an emulsion is generally a blend of crystals, you get basically 3 discontinuous functions, a toe, a straight line portion, and a shoulder. These, taken together represent a cubic spline. And, all films are characterized by having one. In fact, digital uses a V (voltage) Log E curve in much the same way to develop the characteristic curve of the sensors.

So, the cubic spline turns out to be fundamental to the exposure density curve by the fundamental response of the silver halide grains themselves. I'm not sure what the V Log E curve is repsonding to but it can be used in copy machines and digital cameras both to make the same sort of estimate of response to light, but I assum it is all related to low light levels, "normal" light levels, and very bright light levels.

They eye, of course has a defined linear response because we are the observer and the eye can self compensate for these factors and it also integrates the scene.

AAMOF, since a print has a non-linear curve, and the eye is "linear", the curve of the print must be increased wrt the original scene to compensate for the toe and shoulder. So, a print with a mid scale contrast of about 1.5 - 1.7 appears "normal" to the eye as if it had a contrast of about 1.0. The range is determined by the toe and shoulder.

Basically then, one might say that the response of light sensitive materials is non-linear, and may be broken into three types, and these devolve into a good fit for a cubic spline.

PE

gainer · Jul 24, 2008

Amen! Such empirical curve fitting is quite useful when it is necessary to integrate a curve for which you have no equation. It can even be used on random noise, although with current scanning and computing technology it is hardly necessary to do so. When did a practicing photographer ever become concerned with fitting a characteristic curve's shape, other than to recognize that it has a toe, and to use experience gained from failure to place the shadow accordingly? Once that is done, the same meter or practiced eye, aimed at the highlight, will allow her to judge the contrast index needed to allow the negative to fit some available paper's range. The paper also has curvature of the spline, which often compensates at least to some extent that of the film.

gainer · Jul 24, 2008

PE, each film-developer combination IS treated separately. I use two different time-CI measurements at the same temperature to estimate the maximum CI and another at a different temperature to estimate the change of slope with temperature. Besides this explanation, I have to point out that you are claiming that what I did in fact cannot be done.

Photo Engineer · Jul 24, 2008

gainer said:
PE, each film-developer combination IS treated separately. I use two different time-CI measurements at the same temperature to estimate the maximum CI and another at a different temperature to estimate the change of slope with temperature. Besides this explanation, I have to point out that you are claiming that what I did in fact cannot be done.

Patrick;

I am agreeing with you, I think. This is getting rather esoteric and you use terms and methodology different than what we did.

I am saying that no two film/developer combinations can have the same response curve to temperature. A given film in several developers might have a similar response curve, but offset by the activity of thedeveloper, and a given developer might have a similar response curve to temperature when used with different films, but offset by the nature of the films.

And, I'm saying that this is due to the kinetic and thermodynamic properties of the given films and chemicals in the developers.

PE

ic-racer · Jul 24, 2008

Photo Engineer said:
#4 is a big change from what you seemed to be saying before.

Somehow one of use or both, misinterpreted something. In the end I think we are in agreement.

Looks like this thread has taken quite a turn, however. Since we are talking about cubic splines now I just wonder if anyone out there has a good way to solve them (that is easier than looking at a graph).

That is, when seeking 0.1 on the toe for an 'in camera' film EI determination, it has always been easier for me to just look at the plot of the spline on the computer and approximate the EI at the 0.1 crossing point. When I try to get the equation and solve for 0.1 it seems to take 10 times longer. I just wonder if anyone had good software that EASILY plot and solve the cubic spline for 0.1?

Photo Engineer · Jul 24, 2008

Actually, there are 3 methods that can be used here:

1. Develop to a given straight line slope and draw a line along the straight line to intersect the x axis. This log E value is then the "speed" and you can arbitrarily assign it based on the ISO you used in exposure.

2. Draw a straight line along the dmin of the film and where the curve of the film deviates and begins its upward curve of the toe, that is the "speed".

3. Use 1 & 2 above and where these two lines intersect, draw a line perpendicular to the orizontal line down to the X axis. This then becomes the speed point.

As for curve fitting software, all I can say is that all EK software for reading densities automatically fit them to the cubic spline that was derived from the mathematics gurus work.

PE

Developer Temperature Compensation Formula

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