  Cooperative adsorption of gases by porous frameworks permits more efficient
uptake and removal than does the more usual non-cooperative (Langmuir-type)
adsorption. Cooperativity, signaled by a step-like isotherm, is usually
attributed to a phase transition of the framework. However, the class of
metal-organic frameworks mmen-M$_2$(dobpdc) exhibit cooperative adsorption of
CO2 but show no evidence of a phase transition. Here we show how cooperativity
emerges in these frameworks in the absence of a phase transition. We use a
combination of quantum and statistical mechanics to show that cooperativity
results from a sharp but finite increase, with pressure, of the mean length of
chains of CO2 molecules that polymerize within the framework. Our study
provides microscopic understanding of the emergent features of cooperative
binding, including the position, slope and height of the isotherm step, and
indicates how to optimize gas storage and separation in these materials.
