Its density is high and the area it is stored in ( ie. the refill ) is very small

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   Due to the pressure outside the pen is a little more than pressure the inside.  The gravitation and the capillary action allow the ink to come out slowly through the small opening for the nib.
   Air has to enter inside the refill or the ink tank, for some ink to flow out.  There is an arrangement for very little air to go in.  So only very little ink flows out on paper as we write.

   If there is a big gap between the nib and the black plastic part below nib in an ink pen (fountain pen), then ink over flows.

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