Delete Cookie when closing the browser

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C#

ASP.NET

Javascript

Ajax

jQuery

I have a website containing three aspx pages and a MasterPage
1) Login.aspx: containing a textbox and button when login I set a cookie with the text in the textbox

<asp:Label ID="lbl_EnterName" runat="server" Text="Enter name:"></asp:Label><br />
    <asp:TextBox ID="txt_UserName" runat="server"></asp:TextBox>
    <br />
    <asp:Button ID="btn_Submit" runat="server" Text="Submit" OnClick="btn_Submit_Click" />

and here's the code behind:

protected void btn_Submit_Click(object sender, EventArgs e)
    {
        HttpCookie UserCookies = new HttpCookie("Username");
        UserCookies.Value = txt_UserName.Text;
        UserCookies.Expires = DateTime.Now.AddDays(1);
        Response.Cookies.Add(UserCookies);
        Response.Redirect("~/default.aspx");
    }

====================
2) Default.aspx: containing a label shows the name in the cookie

<asp:Label ID="lbl_msg" runat="server"></asp:Label>

and here's the code behind:

protected void Page_Load(object sender, EventArgs e)
    {
        HttpCookie username = Request.Cookies.Get("Username");
     
        try
        {
           lbl_msg.Text = "I'm logged in:   " + username.Value;
        }
        catch
        {
            lbl_msg.Text = "Sorry. you're not logged in!";
        }
    }

====================
3) logout.aspx: just to Delete the cookie
here's the code behind

protected void Page_Load(object sender, EventArgs e)
{
    Response.Cookies["Username"].Expires = DateTime.Now.AddDays(-1);
}

====================

I want to run the Logout.aspx when closing the browser to re-login when open the website again (like facebook, gmail, ....)
I tried to use javascript code in the masterpage but it didn't work for me (events on the body tag -onunload- -onbeforeunload-) .. Thanks in advance