We investigate the enrichment in elements produced by the slow neutron-capture process ($s$-process) in the globular clusters M4 (NGC 6121) and M22 (NGC 6656). Stars in M4 have homogeneous abundances of Fe and neutron-capture elements, but the entire cluster is enhanced in $s$-process elements (Sr, Y, Ba, Pb) relative to other clusters with a similar metallicity. In M22, two stellar groups exhibit different abundances of Fe and $s$-process elements. By subtracting the mean abundances of $s$-poor from $s$-rich stars, we derive $s$-process residuals or empirical $s$-process distributions for M4 and M22. We find that the $s$-process distribution in M22 is more weighted toward the heavy $s$-peak (Ba, La, Ce) and Pb than M4, which has been enriched mostly with light $s$-peak elements (Sr, Y, Zr). We construct simple chemical evolution models using yields from massive star models that include rotation, which dramatically increases $s$-process production at low metallicity. We show that our massive star models with rotation rates of up to 50% of the critical (break-up) velocity and changes to the preferred $^{17}$O($alpha$,$gamma$)$^{21}$Ne rate produce insufficient heavy $s$-elements and Pb to match the empirical distributions. For models that incorporate AGB yields, we find that intermediate-mass yields (with a $^{22}$Ne neutron source) alone do not reproduce the light-to-heavy $s$-element ratios for M4 and M22, and that a small contribution from models with a $^{13}$C pocket is required. With our assumption that $^{13}$C pockets form for initial masses below a transition range between 3.0 and 3.5 M$_odot$, we match the light-to-heavy s-element ratio in the s-process residual of M22 and predict a minimum enrichment timescale of between 240 and 360 Myr. Our predicted value is consistent with the 300 Myr upper limit age difference between the two groups derived from isochrone fitting.
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