I saw in multiple places at FCOM that roll rates are increased while using speed brakes. Can anybody explain why it is so aerodynamically? I looked at b737.org. There is explanation but I cannot understand it. I don't see any difference between photo 2 and 4
Why it was done so? Was it impossible to design speed brakes without roll rate changes?
Not quite sure what is bothering you however a couple of things could be involved.
Looking at your bottom picture it would appear to me that the turbulent flow that would be formed by the airbrake being extended could easily have some effect on the nearby aileron.
Also the basic roll damping of the wing could be changed by airbrake extension. The less efficient a wing is at producing lift is likely to reduce its natural roll damping. (Roll damping being due to a down going wing producing more lift thanks to the AoA increase and the upgoing one less lift - which naturally damps the roll).
Given that airbrakes are only used infrequently I would not expect the designers to compromise the basic aileron/roll rate design isues in order to counter an airbrake effect.
With full speed brake, the roll rate will also be higher, it is not limited to the partial speed brake condition.
In the partial case, the opposite wing spoiler drop couples with the roll side spoiler rise, giving an increased rate, as well as the increased loading of the outer wing for a given condition, which increases the roll response of the ailerons alone. Yes, the AoA is generally higher with the speed brake up, and that acts to mitigate the aileron effectiveness to a point, but mainly at the very high AoA case approaching stall, not at the normal AoA's in operation.
With full speedbrake, the aileron effectiveness alone increases roll rate, if the AoA of the outer section is below stall AoA.
The above is not necessarily true for the B777, 787 or the Airbus, where the controls may change the scheduling of the spoilers and the aileron deflection for a given operating condition. Having said that, the B777 roll rate is increased with spoilers up... On the A320, recall that the roll scheduling was done by the position of the flap handle, and if the TE flaps were stuck at full flap, then repositioning the flap handle would lead to divergence with or without the autopilot engaged.... due to the excessive roll sensitivity from the spoiler deflection with a full flap position. (HDA323 - OEB117)
If I remember correctly on the 727, no speed brakes equals normal roll rate. Half speedbrake equals very high roll rate. Full speed brake equals a roll rate similar to the roll rate with no speedbrake extension(but maybe not exactly the same...I don't know).
Flight spoilers assist the ailerons in rolling the aircraft. Flight spoilers deploy as the control wheel is displaced past 11 degrees to reach a maximum deflection when past 70 degrees. They also deploy but symetrically when used via the speed brake lever,at the flight detent,the flight spoilers are at their full deflection of 20 deg and 22.5 deg for the inboard and outboard respectively.
So if in turn with the wheel displacement increasing past the 11 degrees and the spoilers in flight detent,the roll will not be increased as the spoilers would already be in their full deployed via the speed brake.
In any position other than flight detent,an increase in bank( wheel displacement more than 11 deg) will end up with more up spoilers on one side than the other. If the lever is increased fast and especially with flaps ,overbanking may occurr and or AP disconnect.