Effect of confinement on the phase transitions

Confinement in pores affects all phase transitions of fluids, including the liquid-solid phase transitions (see Ref. [276, 277] for review) and liquid-vapor phase transitions (see Refs. [28, 278] for review). Below we consider the main theoretical expectations and experimental results concerning the effect of confinement on the liquid-vapor transition. Two typical situations for confined fluids may be distinguished fluids in open pores and fluids in closed pores. In an open pore, a confined fluid is in equilibrium with a bulk fluid, so it has the same temperature and chemical potential. Being in equilibrium with a bulk fluid, fluid in open pore may exist in a vapor or in a liquid one-phase state, depending on the fluid-wall interaction and pore size. For example, it may be a liquid when the bulk fluid is a vapor (capillary condensation) or it may be a vapor when the bulk fluid is a liquid (capillary evaporation). Only one particular value of the chemical potential of bulk fluid provides a two-phase state of confined fluid. We consider phase transions of water in open pores in Section 4.3. 

Liquid-vapor phase transitions of confined fluids were extensively studied both by experimental and computer simulation methods. In experiments, the phase transitions of confined fluids appear as a rapid change in the mass adsorbed along adsorption isotherms, isochores, and isobars or as heat capacity peaks, maxima in light scattering intensity, etc. (see Refs. [28, 278] for review). A sharp vapor-liquid phase transition was experimentally observed in various porous media ordered mesoporous sifica materials, which contain non-interconnected uniform cylindrical pores with radii Rp from 10 A to more than 110 A [279-287], porous glasses that contain interconnected cylindrical pores with pore radii of about 10 to 10 A [288-293], silica aerogels with disordered structure and wide distribution of pore sizes from 10 to 10 A [294-297], porous carbon [288], carbon nanotubes [298], etc. 

It is very difficult to measure the coexistence curves of confined fluid experimentally, as this requires estimation of the densities of the coexisting phases at various temperatures. Therefore, only a few experimental liquid-vapor coexistence curves of fluids in pores were constructed [279, 284,292,294-297]. In some experimental studies, the shift of the liquid-vapor critical temperature was estimated without reconstruction of the coexistence curve [281-283, 289]. The measurement of adsorption in pores is usually accompanied by a pronounced adsorption-desorption hysteresis. The hysteresis loop shrinks with increasing temperature and disappears at the so-called hysteresis critical temperature Teh. Hysteresis indicates nonequilibrium phase behavior due to the occurrence of metastable states, which should disappear in equilibrium state, but the time of equilibration may be very long. The microscopic origin of this phenomenon and its relation to the pore structure is still an area of discussion. In disordered porous systems, hysteresis may be observed even without phase transition up to hysteresis critical temperature Teh 7c, if the latter exists [299]. In single uniform pores, Teh is expected to be equal to [300] or below [281-283] the critical temperature. Although a number of experimentally determined values of Teh and a few the so-called hysteresis coexistence curves are available in the literature, hysteresis 

Phase transitions of confined fluids were extensively studied by various theoretical approaches and by computer simulations (see Refs. [28, 278] for review). The modification of the fluid phase diagrams in confinement was extensively studied theoretically for two main classes of porous media single pores (stit-Uke and cylindrical) and disordered porous systems. In a slit-like pore, there are true phase transitions that assume coexistence of infinite phases. Accordingly, the liquid-vapor critical point is a true critical point, which belongs to the universality class of 2D Ising model. Asymptotically close to the pore critical point, the coexistence curve in slit pore is characterized by the critical exponent of the order parameter = 0.125. The crossover from 3D critical behavior at low temperature to the 2D critical behavior near the critical point occurs when the 3D correlation length becomes comparable with the pore width i/p. 

It is not clear how two phases coexist in disordered pores as alternating domains or as two infinite networks. Disordered porous materials with low porosity are more reminiscent of interconnected cylindrical pores and therefore a domain structure seems to be more probable [299, 311-315]. In highly porous materials, such as highly porous aerogels, infinite networks of two coexisting phases may be assumed. The critical point of fluids in disordered pores is expected to belong to the universality class of the random-field Ising model [316-318]. 

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