Hello Spectral532,
Welcome to CPF.
As others have mentioned this is a great way to charge Li-Ion cells.
However there is one "issue." Li-Ion chargers will terminate the charge when the charge current drops below 50 - 100 mA. The reason they do this is because Li-Ion chemistry does not handle "trickle charging" very well.
In this case, we are defining "trickle charging" as the charge that occurs when the cell is at 4.2 volts and the current is in the range of 50 - 0 mA.
If you let your power supply go, it will continue to charge until the current drops to 0 mA. The "trickle charging" phase tends to oxidize the electrolyte inside the cell, and is detrimental to long term life. It also takes a long time to complete this final stage, so charging time is extended. This is especially true when the cell has developed some internal resistance and is no longer holding 4.2 volts rested. With a cell that has been in service for awhile and is at around 90% of its original capacity, the amount of time spent "trickle charging" can be equal to or exceed the total charge time if the charge was terminated when the current dropped to around 100 mA. As the cell ages further, you get into the situation where the "trickle charge" never ends.
There are a couple of ways to work around this.
One way is to be very vigilant and stop the charge when the current falls below a certain amount. This is a very good way to go, but sometimes we get distracted and let things go a little longer than we had planned.
A better way is to simply reduce the voltage you are charging to.
Cycle life is related to the maximum voltage you charge the cell to. If you charge to 4.2 volts, you should get somewhere around 500 charge/discharge cycles. However, if you limit the charge to 4.1 volts, the cycle life extends to somewhere around 1500 charge/discharge cycles.
With this information in hand, you can now adjust the settings to avoid any problems with "trickle charging." You simply set the voltage to 4.15 or 4.18 volts. With cells in good condition I usually use 4.18 volts. As the cells age and develop some internal resistance, I drop that to 4.15 volts. Many people simply use 4.15 volts all the time.
With this method you still have to watch the charge process, but you don't have to be glued to it. Since our definition of "trickle charge" is very specific, at a lower voltage we are not "trickle charging." Charging using a power supply is safest using this method.
There is one other method that can be used. Normal charge rates are in the 0.5C - 1.0C range. An optimum rate is about 0.7C. The idea is to balance the CC phase and the CV phase of the charge. The faster you charge, the more time you spend in the CV phase, and the greater the possibility of "trickle charging." The ideal rate (from the cells perspective) is 0.7C, but everyone wants faster charging and I am not sure there is much of a difference in cell life between charging at 0.7C or 1.0C.
At any rate, if you charge at a low charge rate, it seems that you can eliminate the CV stage completely. I have not tried this, but have read that if your charge current is about 0.18C, or lower, you can simply terminate the charge when the voltage reaches 4.2 volts.
Once again you have to watch the end of the charge closely, but it is another option. The general rule that the termination current is usually about 10% of the initial charge current does not apply in this case, but you simply terminate upon the voltage reaching 4.2 volts.
One final caution... from experience.
If you have curious cats, watch them carefully while charging. I had a cat become interested in the changing displays and was pawing at the control nobs...
Tom