The choices of conditions for this reaction are interesting. It is reasonable to assume that DMSO was chosen as the solvent in order to dissolve the highly polar starting material. Cesium carbonate was likely chosen as the base since it should be strong enough to deprotonate the more acidic N-H of the starting material while not containing a nucleophilic anionic base (such as hydroxide) that could potentially hydrolyze the phosphate ester or the chloromethyl group.
Under these conditions, a phase-transfer catalyst may not be needed and the tetrabutylammonium iodide may simply have served as a source for iodide catalysis to form the active iodomethyl group in situ which might be the reason that reaction could be performed at room temperature.
If we were developing a process for this reaction, we would screen the use of the much less expensive potassium carbonate as the base together with 5 mole% tetrabutylammonium bromide as a true phase-transfer catalyst and maybe 1 mole% KI in order to minimize cost. Thermodynamics dictates that TBAB and KI will preferentially form TBAI in-situ instead of buying expensive TBAI. We would expect the potassium carbonate to be sufficiently basic to deprotonate the NH at the interface, plus act as a desiccant, then let the quaternary ammonium phase-transfer catalyst complete the N-alkylation of the N-anion likely formed at the solid K2CO3-liquid DMSO interface.
We do not know the conversion of the reaction, though it is likely very high due to the reported weight of the crude product before chromatography. The low 27% isolated yield may be due to the relatively nonpolar solvent chosen to elute the polar product from the chromatographic column. We also do not know if chromatography might have been needed to separate the possible N-alkylated product at the other NH group of the starting material or even di-N-alkylated byproduct.
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