The thumbnail image is the Garrett collection complete 1868 proof set from the cent to the double eagle struck in aluminum. The image below is of the Heritage 97 ANA set.
These were struck originally for presentation purposes as Pollock mentions a request by Henry Linderman to have "two sets of impressions of our coins struck in Aluminum prepared as early as they can be without an interruption to the regular business of the Mint. I desire to send one set to the Bank of England and another to France". Pollock also notes that Linderman asked for 2 more sets 2 days later. H.R. Linderman received the request below for 5 more sets on June 6, 1868.
"Order received June 6th, 1868 243 from the Director of the Mint by A. Tracker Cek
5 Aluminum sets of the coins of the United States Delivered on or before July 6, 1868 (4 sets as above, were previously struck and delivered on an order from the Treasury Dept at Washington.)"
It is likely that some of these were deliberately made for sale to collectors. The first recorded sale of one of these cased sets was lot 1381 of the 1870 Mason & Company sale of the William Fewsmith collection. This set was bought in by Mason and was offered piecemeal in his next sale but did not sell.
The Brand collection had two sets, Woodside's purchased in 1895 and Stickney's purchased in 1907. The sales catalogs do not make note of these having the case, however.
1) R.A. Britton (H.G. Sampson, 2/1882), lot 956; T. Harrison Garrett; Garrett Collection; Johns Hopkins University (Bowers and Ruddy, 11/1979), lot 396; Superior 10/89 Heifetz sale, Heritage 1/13 FUN.
2) Newcomer?, offered for $850 in Kosoff & Kreisberg's May 1950 edition of Numismatic Gallery Monthly, Bowers and Merena 1/95, Heritage 97 ANA, Heritage 1/07, M. Hagen.
3) Treasury Secretary Hugh McCulloch, McCulloch family, Newman-EPNNES, Heritage 11/14, regrettably sold piecemeal.
Similar aluminum sets were also made in 1869 although probably not with the case. The Doughty (1891) H.P. Smith (1906) set was purchased by Brand and is no longer believed to be intact.
Off-metal die trial sets in copper, aluminum or nickel, both complete or partial, are known for many years including 1867-1876, 1884 and 1885. They were quite commonly seen in auction sales of the late 1800s and early 1900s. Most appear to have been deliberately made for sale to collectors. The partial 1884 copper set was apparently presented to A.M. Smith, who authored the Mint's visitor guides.
There were 3 copper sets known for 1867, Garrett's ex Charles A. Britton-G Cogan 1/1883, which has since been broken up, and the virtually complete double set stolen from the Iowa State Historical Museum. All 3 sets contained just the with rays nickel J572/P648. Iowa State Historical Museum's website notes: "Die trial pieces of 1867 set in plaque of Iowa. Two one-cent pieces, two two-cent pieces, two three-cent silver pieces, two three-cent nickels, two half-dimes, two five cent with rays on top, two dimes, two quarters, two half-dollars, two silver dollars, two gold dollars, one $2.50 gold piece, two $3,00 gold pieces, two $5.00 gold pieces, one $10.00 gold piece, and two $20.00 gold pieces. NOTE: THIS PLAQUE COULD NOT BE FOUND IN 2006".
For those years where sets were made in multiple metals, the nickel ones are the rarest, followed by aluminum and finally copper being the least rare.
Sets for the years 1863, 1864 and some of 1865 were made circa 1869 and into the early 1870s. They combined quarters, half dollars and silver dollars with the "In God We Trust" reverses on them as used from 1866 onward with novodel (backdated) three cent, half dimes and dimes. The trimes and half dimes are known for having an open 'D' in United and the dimes are known for having a broken 'S' in States. Additional pieces that were struck as part of these sets include the 1863 two cent pieces with large motto J316-318/P381-P383 and 1863L and some 1864L Indian cents. These sets were struck in silver, copper, aluminum and nickel depending upon the year.
Photo courtesy of Superior.