First things first: whenever you write again or otherwise contact your potential advisor, stay polite and respectful. Conduct yourself in the most professional manner you can. While you might feel your correspondent is acting unprofessionally in not responding at a rate that pleases you, asking again in anything that looks like a demanding tone is not going to reflect well on you.
There are many possible reasons why your correspondent hasn't replied to your first email yet, even 10 days down the line. You are probably already quite aware of the possibilities, but here are a few:
- He/she is simply busy. Is it a busy time of year? Is it examination time? Is he/she likely to busy marking/providing feedback. providing lectures?
- Is he/she on holiday? A quick check with the Department/School/Institute administration should answer this.
- There could be personal reasons. You don't want to appear a demanding person when the academic is making preparations for a family member's funeral - for example! Extreme sitation, yes, but not impossible.
With these possibilities in mind, you could write again to your potential advisor, reminding him/her that you are very keen to work together. I do not advise simply re-sending your original email unless you have good reason to believe that the original didn't reach its target - e.g. you have received a mailhost error relating to an undelivered message. I suspect this is not the case. In your follow-up email, refer to your previous email. Resending an email is similar to saying to someone "I SAID....!!!".
In your follow-up email, I would acknowledge that the academic might well be very busy and that you would gratefully appreciate any time the academic could spend talking with you about research opportunities. Offer alternatives - you've hinted at this - such as arranging a meeting or a telephone call at a time convenient for the academic.
At this stage, be reasonably formal. Your spelling and grammar should be as good as you can make it. This indicates that you are taking this correspondence seriously, as should he/she.
It is very easy to appear demanding, annoying or otherwise negative in an email, even if that is not your intention. Turn up the politeness up a few notches, even if you think it's starting to be sickly and ingratiating. In emails, it is extremely easy to appear demanding or accusatory. You don't have the advantage of being present in a conversation to send all those little non-verbal clues which make up the vast majority of human communication.
I think that it is appropriate to mention that you have some expectations when you'd like to have this interaction. You could mention that you are meeting with other advisers (if true) and that you'd like to be in a position to make a decision in one/two month's time. For this reason, you'd like to have a meeting sometime in the next 1-2 weeks, if possible.
- You be unfailingly polite and professional,
- You put effort into your next email - don't just resend a previous one,
- You suggest as many alternatives for an interaction you can think of,
- You provide some kind of time-scale for action.