Phase transitions in pores

Over the years, vapour adsorption and condensation in porous materials continue to attract a great deal of attention because of (i) the fundamental physics of low-dimension systems due to confinement and (ii) the practical applications in the field of porous solids characterisation. Particularly, the specific surface area, as in the well-known BET model [I], is obtained from an adsorbed amount of fluid that is assumed to cover uniformly the pore wall of the porous material. From a more fundamental viewpoint, the interest in studying the thickness of the adsorbed film as a function of the pressure (i.e. t = f (P/Po) the so-called t-plot) is linked to the effort in describing the capillary condensation phenomenon i.e. the gas-Fadsorbed film to liquid transition of the confined fluid. Indeed, microscopic and mesoscopic approaches underline the importance of the stability of such a film on the thermodynamical equilibrium of the confined fluid [2-3], In simple pore geometry (slit or cylinder), numerous simulation works and theoretical studies (mainly Density Functional Theory) have shown that the (equilibrium) pressure for the gas/liquid phase transition in pores greater than 8 nm is correctly predicted by the Kelvin equation provided the pore radius Ro is replaced by the core radius of the gas phase i.e. (Ro -1) [4]. Thirty year ago, Saam and Cole [5] proposed that the capillary condensation transition is driven by the instability of the adsorbed film at the surface of an infinite... 

Experimental results on the effect of size on the kinetic behavior of phase transitions in alloys have been obtained in several works [14, 23,85). As an example of size-induced effects and, in particular, size-induced hysteresis phenomena in kinetics, one can mention capillary condensation (phase transitions in pores or capillaries for some recent work in this direction, see [69, 86, 87]). Controllable, continuous, and reversible coexistence of different crystalline and disordered phases in galHum nanoparticles under electron beam excitation has also been demonstrated [88]. Moreover, recent theoretical, experimental, and numerical Monte Carlo results on first-order phase transitions in nanovolumes demonstrate hysteresis phenomena [59, 65, 86, 89-92). 

This methodology developed to observe water freeze-thaw in concrete materials, may be used quite generally to observe solid-liquid phase transitions in many different materials of industrial and technological interest. The method could be also applied to other problems involving freezing and thawing of water in confined pores. 

The results stated so far has been with saturated vapor or liquid as the equilibrium bulk phase. Liquid-like state in pore, however, can hold with reduced vapor pressure in bulk the well-known capillary condensed state. One of the most important feature of the capillary condensation is the liquid s pressure Young-Laplace effect of the curved surface of the capillary-condensed liquid will pull up the liquid and reduce its pressure, which can easily reach down to a negative value. In the section 2 we modeled the elevated freezing point as a result of increased pressure caused by the compression by the excess potential. An extension of this concept will lead to an expectation that the capillary-condensed liquid, or liquid under tensile condition, must be accompanied with depressed freezing temperature compared with that under saturated vapor. Then, even at a constant temperature, a reduction in equilibrium vapor pressure would cause phase transition. In the following another simulation study will show this behavior. 

The problem of adsorption hysteresis remains enigmatic after more than fifty years of active use of adsorption method for pore size characterization in mesoporous solids [1-3]. Which branch of the hysteresis loop, adsorption or desorption, should be used for calculations This problem has two aspects. The first is practical pore size distributions calculated from the adsorption and desorption branches are substantially diflferent, and the users of adsorption instruments want to have clear instructions in which situations this or that branch of the isotherm must be employed. The second is fundamental as for now, no theory exists, which can provide a quantitatively accurate description of capillary condensation hysteresis in nanopores. A better understanding of this phenomenon would shed light on peculiarities of phase transitions in confined fluids. 

A characteristic feature associated with pore condensation is the occurrence of sorption hysteresis, i.e pore evaporation occurs usually at a lower p/po compared to the condensation process. The details of this hysteresis loop depend on the thermodynamic state of the pore fluid and on the texture of adsorbents, i.e. the presence of a pore network. An empirical classification of common types of sorption hysteresis, which reflects a widely accepted correlation between the shape of the hysteresis loop and the geometry and texture of the mesoporous adsorbent was published by lUPAC [10]. However, detailed effects of these various factors on the hysteresis loop are not fully understood. In the literature mainly two models are discussed, which both contribute to the understanding of sorption hysteresis [8] (i) single pore model. hysteresis is considered as an intrinsic property of the phase transition in a single pore, reflecting the existence of metastable gas-states, (ii) neiM ork model hysteresis is explained as a consequence of the interconnectivity of a real porous network with a wide distribution of pore sizes. 

In thermoporometry experiments the pore radius is deduced from the measurement of the solidification temperature and the volume of these pores is calculated from the energy involved during the phase transition. The pore radius distribution and the pore surface are then calculated. The pore texture can be described from numerical values (mean pore radius, total pore volume or surface, etc...) or by curves. For example, curves of figure 1 are the cumulative pore volume vs pore radius while curves of figure 2 are the pore radius distributions. Texture modifications are conveniently depicted by the pore size distribution curves. 

Phase Transitions in Confined Systems. Studies on confined systems show that even relatively simple systems display complex phase behaviour, which is sensitive to changes in the potential interactions, structure of the pore and models for the pore walls. Despite these factors, some simulations have been able to model experimentally observed behaviour accurately. 

As we have mentioned in the Introduction, the location of the critical point of the lowest density liquid-liquid transition of real water is unknown and both scenarios (critical point at positive or at negative pressure) can qualitatively explain water anomalies. Recent simulation studies of confined water show the way, how to locate the liquid-liquid critical point of water. Confinement in hydrophobic pores shifts the temperature of the liquid-liquid transition to lower temperatures (at the same pressure), whereas effect of confinement in hydrophilic pores is opposite. If the liquid-liquid critical point in real water is located at positive pressure, in hydrophobic pores it may be shifted to negative pressures. Alternatively, if the liquid-liquid critical point in real water is located at negative pressure, it may be shifted to positive pressures by confinement in hydrophilic pores. Interestingly, that it may be possible in both cases to place the liquid-liquid critical point at the liquid-vapour coexistence curve by tuning the pore hydrophilicity. We expect, that the experiments with confined supercooled water should finally answer the questions, concerning existence of the liquid-liquid phase transition in supercoleed water and its location. 

More recently, Ustinov and coworkers [72, 73] developed a thermodynamic approach based on an equation of state to model the gas adsorption equilibrium over a wide range of pressure. Their model is based on the Bender equation of state, which is a virial-like equation with temperature dependent parameters based on the Benedict-Webb-Rubin equation of state [74]. They employed the model [75, 76] to describe supercritical gas adsorption on activated carbon (Norit Rl) at high temperature, and extended this treatment to subcritical fluid adsorption taking into account the phase transition in elements of the adsorption volume. They argued that parameters such as pore volume and skeleton density can be determined directly from adsorption measurements, while the conventional approach of He expansion at room temperature can lead to erroneous results due to the adsorption of He in narrow micropores of activated carbon. 

Three-dimensional opal-VO based photonic crystals were prepared by the chemical bath deposition technique. The x-ray diffraction and Raman spectroscopy confirm the crystalline perfection of VO2 impregnated into synthetic opal pores. It is shown from the optical reflectivity measurements that the photonic bandgap of the opal-V02 based photonic crystals composite is governed by the phase transition in VO2. The shift of Bragg diffraction spectra under the pulsed (10 ns) illumination of YAG.Nd laser has been also observed in the opal-VO2 composites. 

To conclude, we have synthesized VO2 with a perfect crystal stmcture in opal pores using the chemical bath deposition technique. The parameters of the semiconductor-metal phase transition in the prepared material indicate the presence of a small amount of oxygen defects. We have achieved a controllable and reproducible variation of the PEG properties of the opal-V02 composite and inverted VO2 composite during heating and cooling. This is due to the change in the dielectric constant of VO2 at the phase transition. We demonstrated dynamical tuning of the PEG position in synthetic opals filled with VO2 imder laser pulses. 

In conclusion, it should be also said that the origin of the hysteresis loop of the adsorption—desorption isotherms of porous polymers is stiU debated and can be interpreted in different ways. For example, there exists an opinion that hysteresis is not related to traditional capillary condensation in the pores, but may be a consequence of the out-of-equihbrium character of phase transitions in real disordered mesoporous polymers [255]. A failure to reach equilibrium under the given experimental conditions may be caused by the slow diffusion rate of the sorbate [256] or slow swelling of the polymeric sorbent on adsorption and slow relaxation of its swollen structure on desorption. Quite often, a subsequent adsorption on the same material results in larger adsorption capacity values. It is the so-called conditioning effect [256] that may imply a nonequihbrium character of the process. Even the reproducibihty of the shape and location of a hysteresis loop of the isotherms may indicate the estabhshment of fast... 

Liquid-liquid phase transition in confined mixtures have been well described by Evans and Marconi (1987). If pf and are the chemical potentials for component 2 of the coexisting phases a and p in the bulk mixture and in the pore, respectively, and if the a phase is the one rich in 2, then... 

The density functional theory appears to be a powerfid tool for studying adsorption on heterogeneous surfaces. In particular, the valuable results have been obtained for adsorption in pores with chemically stractured walls. The interesting, new phenomena, such as bridging, have been discovered in this way. The phase diagrams characterizing various phase transitions in surface layers can be determined quite quickly from the functional density theory. In this context, we emphasize the economy of the computational efforts required for the application of the fimctional density methods. For this reason, the density theory can be under certain conditions, competitive with computer simulations. However, many applications of the density fimctional theory are based on rather crude, oversimplified assumptions, so the conclusions following firom the calculations should be treated very cautiously. 

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