Update: With some experimentation, I think I figured out what was going on. I had an extra 10K resistor in the circuit which wasn't necessary. With both the resistor and the potentiometer in the circuit, I was only able to light up the red channel with the pot turned fully clockwise. Without the extra resistor, a full turn of the pot lit up all the channels and gave me the white coloration I was wanting. The green and blue channels weren't lighting up because of the excessive resistence and the fact that they obviously power up at different levels. I was making the poor assumption that all of the LEDs would power on at the same time despite the fact that green and blue light require different energy levels.