Phase transitions and quantum effects in pore condensates: a path integral Monte Carlo study

Path-integral simulation of solids

The path-integral formulation of the statistical mechanics of quantum many-body systems is described, with the purpose of introducing practical techniques for the simulation of solids. Monte Carlo and molecular dynamics methods for distinguishable quantum particles are presented, with particular attention to the isothermal-isobaric ensemble. Applications of these computational techniques to different types of solids are reviewed, including noble-gas solids (helium and heavier elements), group-IV materials (diamond and elemental semiconductors), and molecular solids (with emphasis on hydrogen and ice). Structural, vibrational, and thermodynamic properties of these materials are discussed. Applications also include point defects in solids (structure and diffusion), as well as nuclear quantum effects in solid surfaces and adsorbates. Different phenomena are discussed, as solid-to-solid and orientational phase transitions, rates of quantum processes, classical-to-quantum crossover, and various finite-temperature anharmonic effects (thermal expansion, isotopic effects, electron-phonon interactions). Nuclear quantum effects are most remarkable in the presence of light atoms, so that especial emphasis is laid on solids containing hydrogen as a constituent element or as an impurity.

Rounding of phase transitions in cylindrical pores

Phase transitions of systems confined in long cylindrical pores (capillary condensation, wetting, crystallization, etc.) are intrinsically not sharply defined but rounded. The finite size of the cross section causes destruction of long range order along the pore axis by spontaneous nucleation of domain walls. This rounding is analyzed for two models (Ising or lattice gas and Asakura-Oosawa model for colloid-polymer mixtures) by Monte Carlo simulations and interpreted by a phenomenological theory. We show that characteristic differences between the behavior of pores of finite length and infinitely long pores occur. In pores of finite length a rounded transition occurs first, from phase coexistence between two states towards a multidomain configuration. A second transition to the axially homogeneous phase follows near pore criticality.

Thermodynamic and Structural Investigations of Condensates of Small Molecules in Mesopores

Liquids and solids consisting of small, mainly van-der-Waals interacting building blocks, such as Ar, Kr, N2, O2, and CO, are among the most simple systems of condensed matter imaginable. As we shall demonstrate in this microreview on our work sponsored within the Sonderforschungsbereich 277, these cryoliquids condensed in mesoporous hosts with typical mean pore diameters of 7 to 10nm are also particularly suitable for the investigation of fundamental questions regarding the thermodynamics and structure of spatially mesoscale confined systems. An exploration of phase transitions like the vapour–liquid (capillary condensation), the vapour–solid (capillary sublimation), the liquid–solid (freezing and melting) and some solid–solid transformations of such pore condensates reveals a remarkably rich, sometimes perplexing phenomenology. We will show, however, that by experiments combining sorption isotherm, X-ray and neutron diffraction, calorimetric and optical transmission measurements, and by referring to concepts, intermediate between surface and bulk physics, a deeper understanding of the mesoscale mechanisms ultimatively responsible for this complex behaviour can indeed be accomplished, both on a qualitative and a quantitative level.

Ordering of hard spheres inside hard cylindrical pores

Isothermal-isobaric simulations on the ordering behavior of hard spheres upon confinement are presented. The radii of the confining cylinders go from 1.1 to 2 in units of the diameters of the hard spheres adsorbed. In all the range of pressures considered the spheres were located in concentric layers, as many as the radius of the hard cylinder would permit. When the pressure increases, the hard spheres go from being loosely arranged to the formation of ordered structures. This change is marked in all cases by a distinct break in the density of spheres in a narrow pressure range. When the tube radius is smaller than 1.5, the high-pressure ordering is determined by the number of coplanar spheres you can have within a circle of radius equal to that of the confining tube. For wider tubes, the change upon compression is determined by the formation of defected two-dimensional triangular lattices wrapped to fit inside the particular cylinder we are considering.

Impact of the carbon pore size and topology on the equilibrium quantum sieving of hydrogen isotopes at zero coverage and finite pressures

Carbonaceous slit-shaped and square-shaped pores efficiently differentiate adsorbed hydrogen isotopes at 77 and 33 K. Extensive path integral Monte Carlo simulations revealed that the square-shaped carbon pores enhanced the selectivity of deuterium over hydrogen in comparison to equivalent slit-shaped carbon pores at zero coverage as well as at finite pressures (i.e. quantum sieving of hydrogen isotopes is pore-topology-dependent). We show that this enhancement of the D(2)/H(2) equilibrium selectivity results from larger localization of hydrogen isotopes in square-shaped pores. The operating pressures for efficient quantum sieving of hydrogen isotopes are strongly dependent on the topology as well as on the size of the carbon pores. However, for both considered carbon pore topologies the highest D(2)/H(2) separation factor is observed at zero-coverage limit. Depending on carbon pore size and topology we predicted monotonic decreasing and non-monotonic shape of the D(2)/H(2) equilibrium selectivity at finite pressures. For both kinds of carbonaceous pores of molecular sizes we predict high compression of hydrogen isotopes at 77 and 33 K (for example, the pore density of compressed hydrogen isotopes at 77 K and 0.25 MPa in a square-shaped carbon pore of size 2.6 Å exceeds 60 mmol cm(-3); for comparison, the liquid density of para-H(2) at 30 K and 30 MPa is 42 mmol cm(-3)). Finally, by direct comparison of simulation results with experimental data it is explained why 'ordinary' carbonaceous materials are not efficient quantum sieves.

Melting and freezing of argon in a granular packing of linear mesopore arrays

Freezing and melting of Ar condensed in a granular packing of template-grown arrays of linear mesopores (SBA-15, mean pore diameter 8 nm) has been studied by specific heat measurements C as a function of fractional filling of the pores. While interfacial melting leads to a single melting peak in C, homogeneous and heterogeneous freezing along with a delayering transition for partial fillings of the pores result in a complex freezing mechanism explainable only by a consideration of regular adsorption sites (in the cylindrical mesopores) and irregular adsorption sites (in niches of the rough external surfaces of the grains and at points of mutual contact of the powder grains). The tensile pressure release upon reaching bulk-liquid-vapor coexistence quantitatively accounts for an upward shift of the melting and freezing temperature observed while overfilling the mesopores.

Thermodynamics of hydrogen adsorption in slit-like carbon nanopores at 77 K. Classical versus path-integral Monte Carlo simulations

Hydrogen in slit-like carbon nanopores at 77 K represents a quantum fluid in strong confinement. We have used path-integral grand canonical Monte Carlo and classical grand canonical Monte Carlo simulations for the investigation of the "quantumness" of hydrogen at 77 K adsorbed in slit-like carbon nanopores up to 1 MPa. We find that classical simulations overpredict the hydrogen uptake in carbon nanopores due to neglect of the quantum delocalization. Such disagreement of both simulation methods depends on the slit-like carbon pore size. However, the differences between the final uptakes of hydrogen computed from both classical and quantum simulations are not large due to a similar effective size of quantum/classical hydrogen molecules in carbon nanospaces. For both types of molecular simulations, the volumetric density of stored energy in optimal carbon nanopores exceeds 6.4 MJ dm(-3) (i.e., 45 kg m(-3); Department of Energy target for 2010). In contrast to the hydrogen adsorption isotherms, we found a large reduction of isosteric enthalpy of adsorption computed from the quantum Feynman's path-integral simulations in comparison to the classical values at 77 K and pressures up to 1 MPa. Depression of the quantum isosteric enthalpy of adsorption depends on the slit-like carbon pore size. For the narrow pores (pore width H in [0.59-0.7] nm), the reduction of the quantum isosteric enthalpy of adsorption at zero coverage is around 50% in comparison to the classical one. We observed new phenomena called, by us, the quantum confinement-inducing polymer shrinking. In carbon nanospaces, the quantum cyclic polymers shrink, in comparison to its bulk-phase counterpart, due to a strong confinement effect. At considered storage conditions, this complex phenomenon depends on the size of the slit-like carbon nanopore and the density of hydrogen volumetric energy. For the smallest nanopores and a low density of hydrogen volumetric energy, the reduction of the polymer effective size is the highest, whereas an increase of the pore size and the density of hydrogen volumetric energy causes the polymer swelling up to a value slightly below the one computed from the bulk phase. Quantum confinement-inducing polymer shrinking is of great importance for realizing the potential of quantum molecular sieves.

Freezing of hard spheres confined in narrow cylindrical pores

Monte Carlo simulations for the equation of state and phase behavior of hard spheres confined inside very narrow hard tubes are presented. For pores whose radii are greater than 1.1 hard sphere diameters, a sudden change in the density and the microscopic structure of the fluid is neatly observed, indicating the onset of freezing. In the high-density structure the particles rearrange in such a way that groups of three particles fit in sections across the pore.

Capillary phase transitions of linear and branched alkanes in carbon nanotubes from molecular simulation

Capillary phase transitions of linear (from C(1) to C(12)) and branched (C(5) isomers) alkanes in single-walled carbon nanotubes have been investigated using the gauge-cell Monte Carlo simulation. The isotherm at a supercritical temperature increases monotonically with chemical potential and coincides with that from the traditional grand canonical Monte Carlo simulation, whereas the isotherm at a subcritical temperature exhibits a sigmoid van der Waals loop including stable, metastable, and unstable regions. Along this loop, the coexisting phases are determined using an Maxwell equal-area construction. A generic confinement effect is found that reduces the saturation chemical potential, lowers the critical temperature, increases the critical density, and shrinks the phase envelope. The effect is greater in a smaller diameter nanotube and is greater in a nanotube than in a nanoslit.

Molecular modeling of freezing of simple fluids confined within carbon nanotubes

We report Monte Carlo simulation results for freezing of Lennard-Jones carbon tetrachloride confined within model multiwalled carbon nanotubes of different diameters. The structure and thermodynamic stability of the confined phases, as well as the transition temperatures, were determined from parallel tempering grand canonical Monte Carlo simulations and free-energy calculations. The simulations show that the adsorbate forms concentric molecular layers that solidify into defective quasi-two-dimensional hexagonal crystals. Freezing in such concentric layers occurs via intermediate phases that show remnants of hexatic behavior, similar to the freezing mechanism observed for slit pores in previous works. The adsorbate molecules in the inner regions of the pore also exhibit changes in their properties upon reduction of temperature. The structural changes in the different regions of adsorbate occur at temperatures above or below the bulk freezing point, depending on pore diameter and distance of the adsorbate molecules from the pore wall. The simulations show evidence of a rich phase behavior in confinement; a number of phases, some of them inhomogeneous, were observed for the pore sizes considered. The multiple transition temperatures obtained from the simulations were found to be in good agreement with recent dielectric relaxation spectroscopy experiments for CCl(4) confined within multiwalled carbon nanotubes.

Spatial confinement effect on the atomic structure of solid argon

Molecules confined in nanopores show unusual behavior not seen in bulk systems. The present paper reports on molecular dynamics simulations of unusual freezing behavior in confined Ar. Similar to bulk Ar, liquid Ar confined in pores with a diameter D>15sigma (5.1 nm), where sigma is the diameter of the Ar atom, crystallizes when the cooling rate is lower than a critical value (Qc). We also find that the spatial confinement does not have significant influence on Qc when D>15sigma (5.1 nm). In the pore of 10sigma (3.4 nm) in diameter, on the other hand, the behavior is dramatically changed. Crystalline Ar does not appear inside the pore even when the system is cooled at a rate lower than the Qc in the bulk system by over two orders of magnitude. Instead, amorphous Ar characterized by local icosahedral configurations is formed in the pore. We further find that, even when crystalline Ar is formed outside the pore, it does not grow deeply into the pore. This supports that the amorphous Ar is actually the most stable phase in the pore. It is well known that Ar is a poor glass former. Our finding that even such an amorphous Ar is the most stable in the pore suggests that, in any system, it is possible to prepare amorphous structure selectively by using nano-molds.

An experimental study of melting of CCl4 in carbon nanotubes

We report dielectric relaxation spectroscopy measurements of the melting point of carbon tetrachloride confined within open-tip multi-walled carbon nanotubes with two different pore diameters, 4.0 and 2.8 nm. In both cases, a single transition temperature well above the bulk melting point was obtained for confined CCl4. These results contrast with what was obtained in our previous measurements using carbon nanotubes with a pore diameter of 5.0 nm, where multiple transition temperatures both above and below the bulk melting point of CCl4 were observed. Our experimental measurements are consistent with our recent molecular simulation results (F. R. Hung, B. Coasne, E. E. Santiso, K. E. Gubbins, F. R. Siperstein and M. Sliwinska-Bartkowiak, J. Chem. Phys., 2005, 122, 144706). Although the simulations overestimate the temperatures in which melting upon confinement occurs, both simulations and experiments suggest that all regions of adsorbate freeze at the same temperature, and that freezing occurs at higher temperatures upon reduction of the pore diameter.

Quantum corrections for the liquid-gas transition of Lennard-Jones particles in two dimensions

The quantum corrections in first-order perturbation theory are semiquantitatively reproduced in the low temperature behavior of the liquid-gas coexistence curve of the simulations-at least for reduced masses down to m(*)=50.