Pressure doesn't grow with volume. The exterior design pressure is constant. The stress on the wall scales linearly with the diameter. Making submarines bigger actually makes it easier because the buoyancy scales cubically with the volume while the weight scales linearly with the perimeter, so the larger the submarine the thicker the walls can be.
There’s several kinds of scaling involved, once the radius increases enough thicker walls are less efficient than internal bracing.
It’s impractical to build something like an Ohio class submarine that can reach the bottom of the Mariana Trench when you also want multiple internal compartments in case of damage.
Internal bracing is to resist buckling. You need it as your cylinder gets longer (as military subs tend to be), but has nothing to do with the diameter. It's also good for torsional strength, which is not really a concern for a pressure vessel but is important for a ship that's going to be in the actual ocean. But for just resisting pressure, once your diameter exceeds 20 times the wall thickness, the relationship is linear.
You can get better efficiency with multiple spherical pressure vessels joined together over a cylindrical vessel, as spherical pressure vessels better distribute the loads than cylinders. This is done in some particularly deep diving military subs, which are then surrounded by an unpressurized cylinder for hydrodynamics.
Even with a spherical sub the diameter impacts a lot of things. For example a large sphere sees significantly lower pressure across the side facing the surface than the side facing the sea floor.
At the depth of the titanic to see a 1% variation in pressure between the top most and bottom most points of a sphere, the sphere would need to be 40 meters in diameter. For context, the pressure vessels of the largest submarine in the world have a diameter of 10.9 meters. Note that pressure at a given depth varies due to things like temperature fluctuations, ocean currents, and even variation in Earth's gravity. Further, the walls of pressure vessels distribute the load - any variations of the pressure get averaged out. It's the same principle as a dome - every element of the sphere is pushing against the adjacent elements and resisting being pushed by those adjacent elements. At the size scale where this is no longer the case, you're not building a pressure vessel. If you're making a dam or a hollow column going down into water, or perhaps a massive dome on the ocean floor, you would need different equations. Even for a submarine you may be concerned with things besides pressure resistance, like collision or sea keeping, as previously stated. But from a pressure resistance standpoint the diameter to wall thickness requirement holds equally true for small exploratory subs and the largest military subs.
A sphere is a great shape for dealing with such forces but it’s just a more complicated system. Rotation can cause metal fatigue, openings get more complicated, etc.
> For context, the pressure vessels of the largest submarine in the world have a diameter of 10.9 meters.
Few subs can reach the titanic at 12,500 feet, at more common crush depths and especially non spherical geometries it’s very much worth considering. Subs often dive and surface at a significant angle.
