  The phase transition of confined fluids in mesoporous materials deviates from
that of bulk fluids due to the interactions with the surrounding heterogeneous
structure. For example, adsorbed fluids in metal-organic-frameworks (MOFs) have
atypical phase characteristics such as capillary condensation and higher-order
phase transitions due to a strong heterogeneous field. Considering a many-body
problem in the presence of a nonuniform external field, we model the host-guest
and guest-guest interactions in MOFs. To solve the three-dimensional Ising
model, we use the mean-field theory to approximate the guest-guest interactions
and Mayer's f-functions to describe the host-guest interactions in a unit cell.
Later, using Hill's theory of nanothermodynamics, we define differential
thermodynamic functions to understand the distribution of intensive properties
and integral thermodynamic functions to explain the phase transition in
confined fluids. The investigation reveals a distinct behavior where fluids
confined in larger pores undergo a discontinuous (first-order) phase
transition, whereas those confined in smaller pores experience a continuous
(higher-order) phase transition. Furthermore, the results indicate that the
free-energy barrier for phase transitions is lower in confined fluids than in
bulk fluids giving rise to a lower condensation pressure relative to the bulk
saturation pressure. Finally, the integral thermodynamic functions are
succinctly presented in the form of a phase diagram, marking an initial step
toward a more practical approach for understanding the phase behavior of
confined fluids.
