You should do concentration dependent nmr measurements and look for changes in chemical shifts, peak integrals/intensities and peakwidths. Additional information can be obtained from dsnamic light scattering.

Maybe you can follow one link from older question:

https://www.researchgate.net/post/How_could_I_measure_the_ability_of_a_polymer_to_produce_micellar_structure

This paper explains how to determine CMC using UV-vis Spectroscopy in details

CMC means that your polymer has amphiphilic characteristics and that micelles are going to be formed in aqueous media ( micellization may also occur In other selective solvents such as alcohols  or even other organic solvents !)

In addition to UV/visible spectroscopy (generally in the presence of a fluorescence probe) the CMC can also be determined by DLS (dynamic light scattering) or surface tension measurements. These experiments have to be carried as a function of the polymer concentration.

you may find a simple way to determine the CMC .

http://pubs.acs.org/doi/abs/10.1021/ie4024403

You could also use conductivity measurements. You make a set of concentrations and when you see a distinct change in the dependence of the conductivity to the concentration plot -conductivity (or surface tension or UV/vis) vs concentration - you are in the CMC area. These are the easy ways compare to DLS that if it is not set up in your lab is more difficult to use.

Conductivity measurement will only work for dissociated polymers.

It dépends  on the concentration range where micellizatin is expected. Below 0.01 moles per li, surface tensionb and conductivity works well, for ionic sytems of course in the case of conductivity. At higher concentration, any thermodynamic property woks well.

I hope NMR not very suitable at very low concentrations. Surface tension and fluorescence are suitable probes reported elsewhere.

Phani Kumar

Of course, it depends on the polymer and the solvent. If you have a clean set of solutions, static light scattering while quite classical might be effective. You will want to measure I above and below the CMC and multiple angles.  You reasonable expect a marked increase in dI/dc as you cross the cmc., so extrapolation of the I(c) curves above and below the transition will give you approximately the CMC as the point where the lines cross.

Clever readers will doubtless suggest paths that will lead to this approach not working.

One question would be whether the polymeric catalyst will self aggregate at all in true solution.

Directly to the original question. If the polymer contains groups adsorbing in the UV range there will be a shift in spectra when passing the cmc. Be aware that you should not measure in the range of absorbance close to values of 2 and higher because spectra will be affected by artefacts.

If polymer itself does not adsorb in the UV range you will have to add tracer molecules adsorbing in the UV range, which will also show a shift in spectra when they get solubilized in less polar environment of micelles formed.

turbiditymeter could be great way to determine cmc for polymers.

Here is a cost-effective method for the determination of CMC using UV-Visible Spectroscopy, kindly read this paper... "Bioreducible polyethylenimine core-shell nanostructures as efficient and non-toxic gene and drug delivery vectors"

DOI: 10.1016/j.bmc.2022.116886