Does a free software for calculating the net charge of amino acids as a function of pH exist?
I like MarvinSketch. You can run it in your browser or on your computer. Under Insert > Template, you can find all the amino acids. Once you've inserted one, select Calculations > Protonation > Isoelectric point and click OK. You'll get a window that looks like this showing the net charge versus pH (I picked a pretty boring one, but other amino acids produce more complex curves):
I came accidentally across this old not fully answered question. Recently I wrote a web application mainly for the calculation of physico-chemical property of proteins (Prot pi). But you can also use this tool to calculate/estimate the titration curve of single amino acids. You can enter the single letter code for your amino acid (e.g. k for Lysine):