twitteroauth is the first PHP Library to support OAuth for Twitter’s REST API. It is developed by Abraham Williams. You can download the source code @ GitHub and the example inside is quite easy to follow.
GitHub – abraham / twitteroauth
 

Unlike the Facebook API, the Twitter API does not support logout. If you want to increase the security, you can force user login every time when he/she clicks the Signin with Twitter button. This can be down by adding the force_login parameter in the GET oauth/authenticate request.
Twitter API – GET oauth/authenticate
 

Append the force_login parameter in the getAuthorizeURL() function.

...
  /**
   * Get the authorize URL
   *
   * @returns a string
   */
  function getAuthorizeURL($token, $sign_in_with_twitter = TRUE) {
    if (is_array($token)) {
      $token = $token['oauth_token'];
    }
    if (empty($sign_in_with_twitter)) {
      //return $this->authorizeURL() . "?oauth_token={$token}";
      return $this->authorizeURL() . "?oauth_token={$token}" . "&force_login=true";
    } else {
      //return $this->authenticateURL() . "?oauth_token={$token}";
      return $this->authenticateURL() . "?oauth_token={$token}" . "&force_login=true";
    }
  }
...

Thanks Abraham Williams for his work on twitteroauth.

Done =)

Reference:

• Twitter API – GET oauth/authenticate
• GitHub – abraham / twitteroauth