Alternative barriers to log barrier for interior-point method

In the interior-point method the log barrier, i.e $-mu log(f_i(x)$ where $f_i(x)$ is one of the inequality constraints of a problem, is commonly used to remove the inequality constraints. I wonder if it is possible to use a different function as barrier? Are there any kind of criteria to build a "custom barrier" given a specific linear (or non-linear) program?