Your ISP may already provide you with access to its SMTP service. Spamblockers that analyze the IP address of incoming email messages may trust emails sent through your ISP more than emails that you send directly yourself. (Especially if your IP address is considered "residential.") Then the problem shifts to convincing your ISP that you're not a spammer, with 1k emails per day.
There are many services that provide SMTP. Alexus provided one, but it's pretty much a commodity, so shop around. These normally charge you based on how many emails you send in a given amount of time.
Another possibility is to get a VPS and set up a mailserver on it. (Taking care to secure it properly. Spammers love these.) Then you're charged for bandwidth and storage space rather than per email. For example, Linode offers clean IP addresses (not blackholed) with hundreds of gigabytes of bandwidth per month. You can send a lot of emails that way for cheap.
Yet another way to go, if your PHP web app is a mailing list management app, is to outsource the mailing list functionality to someplace like Constant Contact. Then you greatly simplify your infrastructure and probably even gain features.