What is the maximum distance for an argentine ant (Linepithema humile) trail? I imagine that there is some point where the ants are unable to make the return trip either through exhaustion or from the chemicals of the trail failing somehow.
I was reading "Adventures among Ants", by Mark Moffett, to see if he said something about the length of the trails of Argentine ants and I was about to give up and give a general answer based on the other information about Argentine ants in the book, when I hit upon this quote by "Physicist turned biologist Jean-louis Deneubourg":
The Argentine ants' exploratory behavior is exceptional in that they mark continually and explore collectively. Whereas other recruitment trails are constructed between two points (e.g., nest and food), their exploratory trails have no known destination, progressively advancing into the unknown. They rapidly lead new explorers to the frontier between the just explored and the about to be explored zones, avoiding situations where ants will end up exploring the same zone twice, and help returning explorers reach the next directly. A wide corridor of the chemically unmarked area is thus systematically "swept' and marked in a minimum time with maximum economy.
The things to know about Argentine ants are, they form supercolonies in which all ants from the same supercolony can mingle indifferently, regardless of the nest they are from. There is no theoretical limit on the size of such a supercolony; if there is one, it is over thousands of kilometers:
The largest [Argentine ant supercolony] extends from Italy to Spain's Atlantic coast, a distance approaching 2,000 kilometers
This means that the length of an Argentine ant trail is definitely not limited by ant exhaustion or chemical evaporation, depending on what we mean by "trail". If interconnected trails can be counted as a single one you could theoretically have a 2000-km Argentine ant trail that is maintained by ants all along its length. In practice the trail lengths are probably still limited, but by their purpose instead of hard physical limitations. The Deneubourg quote says recruitment trails are built between two specific points, such as a nest and food: this example clearly implies a distance limited by nest density (the trail between a nest and food will be from the nest closest to the food, so if nests are all 10m apart the length of the trail will be 5m at most). This would be true of any trail that has a nest as an endpoint, and what other kind are there?
He also says exploratory trails have no fixed destination, but that doesn't mean they cover unlimited distance either: the exploratory front as described clearly moves forward gradually and deliberately, and as Argentine ants are constantly building nests within the territory of the colony, the exploration front will always be a certain limited distance from the nearest nest.
Then again we could also argue that nest density is itself determined by a physical limit on maximum trail length, which could indeed be ant exhaustion (or ant loss of efficiency at least), risk of the trail being perturbed, chemical evaporation (as in, a longer trail would need more ants to maintain it), etc. I can't speak to that.
Trying to find numbers on Argentine ant nest density I fell upon this paper, which gives more information about ant colony structure and gives the closest thing I think you will find to a definite answer to your question:
In experiments using labeled food, and in a 3-year census of nests and trails, we found that food was shared and nests were linked by trails at distances up to 50 meters.
Note that this is "path distance", i.e. the length of interconnected sets of trails. I couldn't find statements on a single trail, but they show a figure and the longest trail on it is between nests that are about 12m apart.
As an aside, this very interesting paper breaks up the "supercolony" concept a bit; while supercolonies are defined by seeing which ants will attack one another, meaning you can legitimately drop an ant from one end of the colony into the other end and it will go to work unmolested as if it were home, the authors of this paper looked at actual worker movement and food sharing, and found that
Argentine ant populations do not function ecologically as single, large supercolonies, but instead as mosaics of smaller, distinct colonies consisting of groups of interacting nests.
It also says the distribution pattern of nests is seasonal (Argentine ants have very ephemeral nests to begin with); this could give some clue as to what determines nest density and whether physical limits on trail length play a role.