Reduction of Antenna Size for Tiny Devices

I work on the design of self-sufficient wireless sensor nodes (edge length range of 5 .... 15mm). Now I'm looking for the minimal volume of the antenna (the small electronic layer is for this consideration negligible but keep in mind the rest of the volume is a metal block => shielding effects of the battery). Assuming constant receiver sensitivity (e.g. -100dbm), transmitting power (e.g. 0dbm) and a range (e.g. 1m): I know the effect of the carrier frequency wavelength -> higher frequencies allow smaller antennas but an increase from 2,4GHz to 24GHz reduces the range by a factor of 10 due to the higher damping in the air. After decision for the carrier frequency (e.g. 2,4GHz) and a antenna shape (e.g. Miniloop-Antenna) I look for an relationship between antenna efficiency and volume. Are there any analytical formulas for any antenna shapes, where I can calculate an efficiency via gain and volume?

Read any common antenna book (like C.A. Balanis Antenna Theory. Analysis and design.) You can find this formulas in most antennas book. For ex.: for aperture antennas (for uniform currents distribution) You can evaluate the beamwidth by this expression: W=l/D, where:
W - -3dB beamwidth (rad)
l - wavelength
D - aperture diameter

Best regards,
Kit-the-great

It is said that adding an absorber in the antenna can shift the frequency lower and reduce the size in this way