Since you won't have free dodecilsulfate anions in the solution but rather bulky and therefore heavy micells (kind of micells present in your system will depends on temperature and concentration of the surfactant) and sodium cathions I would expect that the the current in the system will be limited by low ionic mobility (which influences the molar ionic conductivity of the electrolite) of bulky DS micells. This might be the reason for the described problem.
Thank you for the much-appreciated answers. Some literature points out that CMC (critical micelle concentration) of SDS in distilled water can be measured to observe the electrical conductivity of the solution because above CMC the added SDS increases the electrical conductivity. But it gives very much sense that above CMC those free SDS ions comes together and create micelles and they are no longer free ions in the solution.
What if I add sodium sulfate in the presence of SDS. Can SDS decrease the Sodium sulfates electrical conductivity?
I am using nanoparticles such as Carbon Black in the solution to increase the temperature of the system at the presence of light ( Halogen light bulbs) and SDS has to be added as dispersant/ stabilizer.