the problem with lubricating this type of chain is to get the lubricant where it needs to function - the internal parts of the chain (pins, rollers etc). this calls for a oil that has good creeping and penetrating properties to get it where you need it and also, because most of the relative movements that need lubrication only make small movements, for good adherence and antiwear properties. you may also want to have good anticorrosion properties to protect the exposed outside components of the chain. finally, because excess lubricant may well drip off into the environment a biodegradable fluid is to be preferred.
most modern chain lubricants nowadays are ester based and are available in a range of viscosities, some contain a diluent that helps in getting it where you want it and then evaporates.
there are various lubricant suppliers that offer a range of specialised chain oils and you would be best served to contact them for advice on this specific application.
another important aspect to consider is the way the lubricant is applied. incidentally it can be applied by a brush, but systems that supply a small amount of lubricant by spraying or brushing usually can prolong chain life considerably.
I would agree with the points made by romke. The most important consideration when lubricating roller chain is making sure the lubricant ends up where it is needed. Specifically the sliding interface at the roller ID/bushing OD and bushing ID/pin OD. These interfaces are not readily accessible, so you need a lubricant with suitable viscosity/creep/penetrating characteristics. A good example is RexOil VSK001. In addition to characteristics described it is also environmentally safe and displaces moisture. In the example described in the OP, the lubricant would be applied at each joint interface between the inner/outer link plates and between the inner link plates/roller end faces. Spraying lubricant four places at each pin joint of the chain would be a tedious and time consuming task, but it would put the lubricant where it's needed and minimize waste.
Just for future reference, there is a type of roller chain called "o-ring chain" that would be a better choice for this type of application. The pin joints are grease-lubed/sealed and require no maintenance.
The application shown has stationary chain. The pins do not rotate because the links do not rotate relative to adjacent links. This is referred to as a pin gear drive.
My opinion is that O-ring chain will not be of benefit in this application because only the pin and i.d. of bushing have a sealed lubricant.
Diamond Chain makes a product known as Dura Lube Chain. I do not think it is available in as large of a pitch as shown in the picture, however.
The vertical bushing makes it difficult to keep lubricant in position. Also, limited roller rotation on the bushing means that the lubricant is not distributed.
For this application a roller pinion drive would be a better arrangement.
Grease may be a suitable lubricant but the grease must meet some specific requirements given your experience so far. Getting dirty when grease lubricated is difficult to avoid because the grease will "catch" all kinds of air borne dust etc which under some conditions may turn into a kind of "grinding paste" accelerating wear. Rust itself need not be a problem with grease lubrication when the grease used has good anti-corrosion properties. From the point of grease lubrication a suitable grease might be a good water repellent grease with anticorrosion additive, like for example a waterfree Ca grease or a Ca-sulphonate based complex grease. However, both greases will collect dirt over time.
