For the Arduino chip or any other uC chip, the signal at the gate will look like an exponential followed by a plateau, followed by another exponential.
I know I wasn't going to post in this thread anymore, but what you mentioned is worth a comment.
The two "exponentials" that you see have different causes.
The first one is due to the Miller effect combined with the inherent gate capacitance of the MOSFET. The "plateau" you see is when the drain is pulled as low as it's going to go, then finally the second exponential is the gate capacitance alone charging to Vgs equilibrium.
I don't think most people really know what the Miller effect is. Anywhere you look it up online, there are all kinds of formulas and hokey-pokey, but no clear explanation of the actual mechanism.
Imagine any 3 terminal (triode) device such as a BJT, a MOSFET and even a vacuum tube triode.
There is inherent capacitance that exists between each terminal and every other terminal, but the one that causes the Miller effect is the drain to gate capacitance (or collector to base, or plate to grid).
Imagine a MOSFET with the drain is being used to switch an LED on through the appropriate resistor.
Initially, the drain is at Vdd (minus the drop across the LED) and the gate is at 0.
Therefore, the inherent capacitor between the drain and gate is charged to Vdd - V-LED.
Now, begin to raise the gate voltage. As the MOSFET begins to turn on, the drain begins to lower, beginning to turn on the LED and also lowering the drain side of the inherent capacitor.
Of course, the gate side of the capacitor will also start to lower, which is FIGHTING the thing trying to drive the gate high.
The gate ends up looking like a LARGER capacitor which takes a lot of current to charge.
Once the MOSFET is saturated (that is, Vgs is at it's minimum), the "capacitance amplifying effect" stops and you get the plateau. Finally, you get another, less "severe" exponential as the gate (and it's inherent capacitance alone) is charged past the Vgs threshold and ends when Vgs is equal to the voltage of the gate driver.
As you can see, if the MOSFET were not connected to a load and the drain instead tied to ground, there would be virtually no Miller effect since the drain can't "push against" the gate. It's already at minimum and won't move any further.
Make sense?