Forgive if I'm rude here.
Design is one of the profesions that has prostituted the most. Why do anyone has to pay a penny if there are always new insecure people who has no previus clients?
"so this would look good on my portfolio"
Your portafolio will look good if it has good work. So why don't you concentrate in making good work to show?
You say you are a designer. But I don't know if you have an idea what design is for. Do you belive in design? Do you belive that a good motion design video will help your client sell more? An impresive web page? A great business card will make a good impression?
"I don't have any clients"
But you don't belive it becouse you are not making any video to sell the idea of making videos. You are not making a web page to sell web page design.
One method to sell is "Look this are my clients". But the best selling argument is "Imagine what I can make for you". There are a lot of people willing to pay talented work. Period.
"By free work, I mean that the clients don't have to pay a single penny for the work that I do."
We know what that means. But new designers have no idea what overall perception that makes! Design is worthless.
I have a question. Does your work worth one penny or less? Or it is good enough to recive real money.
Make a series of wallpapers with an idea of a clasical author, Ilustrate a poem, animate a folkloric tale, make a political statement, make logos out of plain words, make your corporative image and webpage and fool everyone to think is the greatest design firm.
That will get you clients.