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Swelling of clay minerals is an important phenomenon that is relevant to many problems in geoscience, oil and gas reservoirs, geophysics, and engineering. The phenomenon is even more complex when carbon dioxide is also present. Since sequestration of CO2 in sedimentary rock has been under active consideration as a way of alleviating the increasing concerns over climate change for which CO2 is a prime culprit, studying swelling of clays minerals in the presence of CO2 has taken on added urgency. The swelling changes the permeability and porosity of porous formations, which in turn may lead to a reactivation of dormant fractures, triggering seismic activities, and opening up new pathways for CO2 to return to the porous formations' surface. Although the large majority of sedimentary rocks contain various types of mixed-layer clays (MLCs) with intermixed stacking sequences of several types of distinct layers within a single crystal, the vast majority of the past experimental and computational studies of the swelling was focused on pure clays. We present the results of extensive molecular dynamics simulation of hydration and swelling of illite-montmorillonite (I-MMT) MLCs, the most common type of mixed clays, in the presence of both water and CO2 and various combinations of interlayer cations, Na+ and Kt To understand the differences with pure clays, we also report on the same phenomenon in the MMT only. At low CO2 concentrations in Na-MMT, which has its layers' charge concentrated in its octahedral sheet, weak ion surface interactions result in fully hydrated ions and, therefore, more extensive swelling than in I-MMT. Without CO2, adsorption of the cations at the illite surface increases the hydrophobicity of its surface. Thus, in the asymmetric interlayer of I-MMT, illite with stronger surface ion interactions causes accumulation of cations near its surface, which limits their hydration and, therefore, controls swelling of the MLCs. Further inhibition of swelling of Na-MLC can be achieved by increasing the concentration of K+ in the interlayer. At higher CO2 concentrations, however, intercalation of water and CO2 results in a completely different behavior, since in the 2W and 3W hydration states, the hydrophobicity of the MMT surface is stronger than that of illite. Therefore, the density distributions of the intercalated molecules are considerably different from that with water only, which together with disruption of the water network by CO2 reduces the difference between the extent of swelling of the MLC and pure MMT.