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         Photochemistry can be just about as simple or as complex as you want to make it.  When you really get deeply into the science, you're probably going to make things a bit more complicated, since you're most likely to want to synthesize your own developers and fixers (and if you get waaaaay deep into it, color developers, bleaches, and chemical reversal agents).  But the great thing about photochemistry in this day and age is that if all  you want is the basic thrill of developing your own film and printing your own negatives, you can do it with a small financial investment and no more skills as a chemist than you'd probably need to bake a cake from a cake mix.
         Essentially, images form on photographic film and paper because it's coated with some kind of grain (usually a compound of silver and either bromine or chlorine) whose chemical state is moderately altered when it is struck by light.  The grains that have been disturbed will blacken (think: "tarnish") when immersed in a moderately alkaline (and usually organic) solution known as "developer".  The developing action can be arrested either by thoroughly washing the film in water (to remove the developer solution) or by immersing it in a moderately acidic solution known as stop bath.  After this, the film is a mixture of blackened grains and translucent milky white (undeveloped) grains.
         The undeveloped grains must be removed from the film for two reasons.  First, they're milky enough that they don't really pass light (and they tend to diffuse and defocus it), so it's impossible to print through them.  Second, in time they'll tarnish themselves and destroy the negative.  The process of removing the grains is called "fixing" since it makes the image stable or "fixed" -- and therefore durable.  And a great deal of the talk about "archival fixing" has to do with how thoroughly undeveloped grains have been removed.
         After fixing, film and paper need to be washed to remove the fixng agent itself, which is usually caustic in its own right.  And it's not uncommon to hear the term "archival washing" as well as archival fixing, since how well the fixer solution has been washed out of the film can also end up being a determining factor in the negative or print's longevity.  Washing for extended periods of time can sometimes enhance desired effects in certain processes -- notably intensifying the final yellow-green stain characteristic of a pyrocatechin- or pyrogallol-developed negative.  Because it's extremely difficult to "overwash" a negative (at least a B&W negative), if you can afford the water bill, you're probably better off with a long wash.
         So to recap, film development in its most basic form follows the sequence:
Stop Bath or Water Washing
Water Washing
         The sequence can be expanded to also include a "pre-wash" of the film in plain water, and the common reasons for this are to soften the film to make it more receptive to developer, to warm or cool the film to the temperature at which development will proceed, or to wash away a chemically inert layer of the film known as the "antihalation layer"  (The antihalation layer is important in keeping light striking the silver grains from bouncing off one grain and striking another one that it wasn't supposed to hit.  But once the exposure of the film is made, the antihalation layer has no real chemical purpose in the film.  And in some extreme cases -- particularly in certain color reversal films -- failure to thoroughly remove it can result in color balance shifts in the final image.)
         Some processes also call for following fixing with a reimmersion of the film into the exhausted developer in order to enhance a chemical stain in the film.  This is actually safe to do post-fixing because the developer is inert (at least as far as being a developing agent), all those grains that will turn dark already have, and those that won't have already been removed from the film.  Generally speaking, once a fixer has been used on film, it is impossible to make the negative image any darker or denser without running a special process called "intensification" (which is usually accomplished by reimmersing the film in a "chromium intensifier -- a mixture of potassium dichromate and hydrochloric acid) and *then* redeveloping.
         After fixing, some people add a stage called "Hypo Clear," which is designed to remove the last vestiges of the fixer solution.  Frankly, though, this is an unnecessary step provided the water wash is thorough enough.  Hypo clearing agents can also have deleterious effects on certain films and processes -- notably a hypo clearing agent will usually strip a pyrocatechin or pyrogallol negative of its highly desirable yellowish-green stain.  In general I almost never use Hypo Clear in my own work.
         I also vastly prefer to use water washing rather than an acid stop bath for the reason that carbon dioxide (CO2) bubbles can form below the surface of the film when the acidic (usually acetic acid) stop bath hits the alkaline developer.  These bubbles can often cause blemishes on the final negative -- usually manifesting themselves as little clear pinpricks over the surface of the image.  There are rare instances in which I absolutely, positively want to abruptly arrest development; and in those cases I will use a dilute acetic acid stop bath (albeit carefully).  But in most instances I find that there's virtually no difference in density between water-washed and acid-stopped negatives.  And come printing time, any minute difference in density is, pardon the pun, lost in the wash.
         My last wash step I tend to do in distilled water, since I have yet to find a municipal water supply that's clean enough for a final wash.  In Austin, Texas, where I live now, the water supply filters through limestone and leaves considerable quantities of chalky calcium carbonate behind when it evaporates.  I also add a chemical from Kodak known as "Photo-Flo" (and which is made by other manufacturers and marketed under names such as "Anti-Stat," etc.) to that last wash.  Essentially Photo-Flo is nothing more than highly refined liquid dishwashing detergent, and its purpose is to break the surface tension of the last, evaporating water bath by enough that water spots don't form on the negative.
         So for those of you who are curious, my most commonly used B&W photo process looks something like this:
Pre-Soak in Water
Development in a Pyrogallol-Based Developer
Water Wash to Clear the Developer
Fixing in an Ammonium Thiosulfate based fixer
Reimmersion Directly into Spent Pyrogallol Developer Solution
Final Rinse in Distilled Water and Photo-Flo Mixture

         The more common commercially available developers (and their developing times with some of the more commonly available films) are listed below.
         It's worth noting that there are big differences between "lack of grain," "resolution," and "acutance;" and often times developers that deliver strong performance in one category perform dismally in another.
         Lack of grain tends to be the holy grail of 35mm photographers, largely because the film format is small enough that when enlarging enough to make even a reasonable size print, grain becomes noticeable.  In medium and large formats, grain is a lot less critical since the degree of enlargement is a lot less.  So in formats larger than 35mm, grain size can be traded off against other considerations.
         Resolution is a highly measurable quantity that can tell you what the smallest discernable feature of a negative is.
         Acutance is a somewhat subjective "perceived" sharpness, and in fact commonly those film and developer combinations with the highest resolution and the finest grain are not those with the highest acutance.  Kodak's Microdol-X, for example, is known as a developer capable of deliverig extremely fine grain, yet its acutance is terrible when used with most films.
         One of the very few developers capable of delivering *both* high acutance and smooth, fine grain is pyrogallol, and it is for this reason that I do the bulk of my B&W work in PMK Pyro.
         Once a photographer begins working in film sizes greater than 35mm, graininess becomes much less of an issue.  The 6cm x 7cm format I most commonly use when I step up from 35mm produces a negative with about six and a half times the total area as one made on 35mm film.
         Releasing the constraint of keeping grain down allows the photographer to make film and developer selections based on the other criteria of tonality and acutance.  And certain higher speed films with larger grain structure can produce negatives whose tonal range is much better suited to subject.
         One of the more recent advances (and I use the term exceedingly loosely) is the so-called "T-grain film."  T-grains are extremely uniform in both size and shape, and for this reason at any given film speed they will tend to show less grain than a film where grain size and shape is less uniform.  They also are blessed with the property (and it's a mixed-blessing) that they reproduce tones exceedingly linearly.  Their characteristic curves do not tend to have the classic "toe" and "shoulder" of older generation films.
         This comparative lack of toe and knee also works against the photographer in a great many circumstances where there's considerable information in high and low tones of the image and some compression of those tones would actually be quite desirable.
         Kodak T-grain films (T-Max, etc.) tend to have a T-grain "aspect ratio" (the ratio of the width of the grain to its thickness) of about 9:1.  For Ilford T-grain films (Delta, etc.), the ratio drops to about 6:1, and some of the linearity and "toe and knee" characteristics are by-products of these aspect ratios.
         In practical terms, T-Max films are excellent for reprographics -- and not very much else.  I know of very few serious photographers who are actually willing to shoot original negatives onto them.  With a somewhat lower aspect ratio, Ilford T-grain films are a little
         In printing, the sequence of events is roughly the same as it is with film.  Of course, except for the development of large format films, most film developing is done in light-tight tanks, and most paper development (except for certain color processes that are run in light-tight drums) is usually done in trays.  But the general order of steps (Develop - Stop - Fix - Wash - Dry) is about the same.
         The differences are that except for certain unusual printing processes (Sabbatier printing, argentothiocyanate replating, etc.), the "pre-wash" step is almost always omitted.  Stop bath is almost always used in preference to water wash, since developer washed off prints does accumulate in the stop bath tray, and were it full of water, eventually the contents of the tray would be chemically more like a diluted developer than anything that'd arrest development.  Finally, fixing is usually best done in two steps -- a "first fixer" that starts the multi-step process of fixing and a "final fixer" that finishes the process off.  The considerations for doing two-stage fixing are primarily archival, and most home hobbyists are happily oblivious to the fact that something more than a single tray of fixer is ever required.
         Thus, my printing chemistry process (which is highly typical of such a printing process) looks like this most of the time:
Develop in a Metol-Hydroquinone based developer
Stop Bath (Acetic Acid with a pH indicator chemical)
Fixer #1
Fixer #2
Wash in Tap Water
Final Wash in Pure Distilled Water
         Once I am through with a printing session, I usually throw away my first fixer and pour my now slightly-used Fixer #2 into the first fixer bottle.  In this way the second, fresher (second) fixer from the current session becomes the first fixer of the next session; and next printing session I'll mix a brand new batch of second fixer.
         Agitation is an important element in any chemical reaction, and film development is no exception.  Lack of agitation will cause uneven, streaky, or splotchy development.  Too much agitation in a developer can cause socket hole or "surge" marks on a sprocketed film like 35mm or strange patterns at the edges of a roll film or sheet film.  And it's worth noting that required agitation levels can vary widely from developer to developer.  The old standby D-76 requires moderate agitation, but overdoing it will lead to the aforementioned surge marks.  On the other hand, PMK Pyro requires a lot of agitation, and I think nothing of continuously, vigourously shaking the bejeezus out of a tank full of pyro.  A lot depends on what the chemical constituents of a developer are, how quickly they exhaust themselves, what their dilution is, and particularly (in the case of a developer with two developing agents) whether both agents are active or one agent is active and the other is the "replenisher" for the first.
         Developers are not the only chemicals requiring agitiation, although the degree of agitation can vary with the chemical and its role in the process.  Some chemicals such as E6 color reversal bath (which is primarily a comination of acetic acid to stop development and stannous chloride to effect chemical reversal) require only initial agitation to dispese the chemical through the tank, followed by a long, unagitiated soak.  Overagitation of a color reversal bath, although unlikely to produce surge marks, can frequently cause a lowering of color saturation in the final film.  On occasion, such a "pastel" effect can be desirable; however usually it's considered a processing mishap.
         Athough it's not a "chemical" subject per se, since we've touched on a few basic darkroom subjects here, I thought I'd take a moment to discuss safelights -- and just how unsafe they can be.