The annular space around a tubular string has a shape memory material that is in a low profile configuration for run in. After the desired position is obtained and the annulus has cement delivered to fill the annular space, the shape memory device is triggered to revert to an original shape that spans the annulus to seal the tubular and the wellbore sides of the annular space against gas migration through the cement. The structures can have varying run in shapes and can also have original shapes that when the material is triggered will act to displace cement to enhance its compaction on the tubular or the wellbore wall. Combinations of shape memory alloys and polymers are also contemplated to enhance the seal against gas migration.