Several factors play into the relative drop in the African-descended populations of Argentina, but the most important one by far is white European immigration. I'm not exactly sure where OP's 40% figure comes from (although the .4% comes from the 2010 census), and it seems a bit high, although certainly some provinces, especially the more rural ones, did have black populations of that size early in Argentina's history. Assuming that OP is referring to the 1789 pre-independence census- we are starting from a 40% black population out of about 420,000 people. The Viceroyalty of Rio de la Plata was never densely populated, as the region had a relatively small native population and was generally not suited for the kinds of labor intensive plantation agriculture (especially sugar) that resulted in the importation of millions of African slaves to Brazil and the Caribbean.
Anyway, several factors play into the relative drop in the African-descended populations of Argentina, but the most important one by far is white European immigration. The importation of slaves to Argentina was banned in 1824, at which time the population of the country had increased to about 766,000 people. Within a century, by 1914, Argentina's population had increased to over 8 million, and by 1960 Argentina had over 20 million people. This was in large part due to the immigration of millions of Europeans (mostly Spaniards and Italians). But at the same time as the white populations was increasing, the black population had every reason to decrease.
Between the War of Independence, a series of civil wars between the Unitarians and the Federalists, the War of the Triple Alliance against Paraguay, and the War of the Desert against the indigenous peoples of the Pampas, the 19th century was quite bloody for Argentina- and blacks made up large portions of the armies involved in each of these conflicts (and even the majority in San Martin's independence army). This was driven both by the basic fact that people of lower socioeconomic status tend to end up in armies, and because until 1853 (and the abolition of slavery in Argentina), slaveholders were obligated to provide a percentage of their slaves to the military in times of war (who would be freed, if they survived). Unfortunately, many did not survive- both because of the ridiculous numbers of deaths from disease in pre-20th century warfare, and the tendency that black units (as the army was segregated) would often be used as front line cannon fodder. And of course, the deaths of large numbers of young males reduces the ability of an ethnic population to reproduce itself- leaving the women to form mixed-race relationships instead, further whitening the country over time (as individuals with black ancestors would often cease to identify as black/mulato if they were able to pass as white).
Finally, diseases among civilian populations have played a role, in particular a series of yellow fever epidemics in the capital in the 1850s and 1870s. Epidemics tend to hit hardest among the poorest, and most crowded populations. The 1871 epidemic, which killed about 14,000 people started in the San Telmo neighborhood, which had a large black population living in tenements, for example.
All said, blacks were treated as second class citizens even after slavery ended. Many died in war, others in plagues, and those who could left the country (mostly to go to Uruguay). As this population group's growth stagnated at best, millions of Europeans came in and whitewashed Argentina, to the point that now only a fraction of a percent identify as Afro-Argentines. But about ten times that many people believe themselves to have at least one black ancestor, and it is likely that many more have African descent in their family gene pool as well but are unaware.