Luis G. MacDowell

The evaporation/condensation transition of liquid droplets

The condensation of a supersaturated vapor enclosed in a finite system is considered. A phenomenological analysis reveals that the vapor is found to be stable at densities well above coexistence. The system size at which the supersaturated vapor condenses into a droplet is found to be governed by a typical length scale which depends on the coexistence densities, temperature and surface tension. When fluctuations are neglected, the chemical potential is seen to show a discontinuity at an effective spinodal point, where the inhomogeneous state becomes more stable than the homogeneous state. If fluctuations are taken into account, the transition is rounded, but the slope of the chemical potential versus density isotherm develops a discontinuity in the thermodynamic limit. In order to test the theoretical predictions, we perform a simulation study of droplet condensation for a Lennard-Jones fluid and obtain loops in the chemical potential versus density and pressure. By computing probability distributions for the cluster size, chemical potential, and internal energy, we confirm that the effective spinodal point may be identified with the occurrence of a first order phase transition, resulting in the condensation of a droplet. An accurate equation of state is employed in order to estimate the droplet size and the coexisting vapor density and good quantitative agreement with the simulation data is obtained. The results highlight the need of an accurate equation of state data for the Laplace equation to have predictive power.

Tracing the phase diagram of the four-site water potential (TIP4P)

We present here the phase diagram for one of the most popular water models, the four-point transferable intermolecular potential (TIP4P) model. We show that TIP4P model, does indeed provide a qualitatively correct description of the phase diagram of water. The melting line of the five-point transferable intermolecular potential (TIP5P) at low pressures is also presented.

Nucleation and cavitation of spherical, cylindrical, and slablike droplets and bubbles in small systems

Computer simulations are employed to obtain subcritical isotherms of small finite sized systems inside the coexistence region. For all temperatures considered, ranging from the triple point up to the critical point, the isotherms gradually developed a sequence of sharp discontinuities as the system size increased from approximately 8 to approximately 21 molecular diameters. For the smallest system sizes, and more so close to the critical point, the isotherms appeared smooth, resembling the continuous van der Waals loop obtained from extrapolation of an analytic equation of state outside the coexistence region. As the system size was increased, isotherms in the chemical potential-density plane developed first two, then four, and finally six discontinuities. Visual inspection of selected snapshots revealed that the observed discontinuities are related to structural transitions between droplets (on the vapor side) and bubbles (on the liquid side) of spherical, cylindrical, and tetragonal shapes. A capillary drop model was developed to qualitatively rationalize these observations. Analytic results were obtained and found to be in full agreement with the computer simulation results. The analysis shows that the shape of the subcritical isotherms is dictated by a single characteristic volume (or length scale), which depends on the surface tension, compressibility, and coexistence densities. For small reduced system volumes, the model predicts that a homogeneous fluid is stable across the whole coexistence region, thus explaining the continuous van der Waals isotherms observed in the simulations. When the liquid and vapor free energies are described by means of an accurate mean-field equation of state and surface tensions from simulation are employed, the capillary model is found to describe the simulated isotherms accurately, especially for large systems (i.e., larger than about 15 molecular diameters) at low temperature (lower than about 0.85 times the critical temperature). This implies that the Laplace pressure differences can be predicted for drops as small as five molecular diameters, and as few as about 500 molecules. The theoretical study also shows that the extrema or apparent spinodal points of the finite size loops are more closely related to (finite system size) bubble and dew points than to classical spinodals. Our results are of relevance to phase transitions in nanopores and show that first order corrections to nucleation energies in finite closed systems are power laws of the inverse volume.

Can simple models describe the phase diagram of water

The melting point of ice Ih for the TIP3P, SPC, SPC/E, TIP4P, TIP4P/Ew and TIP5P models has been determined by computer simulation. It has been found that the melting points of ice Ih for these models are 146, 190, 215, 232, 245 and 274 K respectively. Thus from the models of water available so far onl yT IP5P reproduces the experimental melting point of water. The relative stability of ice II with respect to ice Ih at the normal melting point has also been considered. Ice II is more stable than ice Ih for the TIP3P, SPC, SPC/E and TIP5P models. Only for the TIP4P and TIP4P/Ew models is ice Ih more stable than ice II at low pressures. The complete phase diagram for the SPC/E, TIP4P and TIP5P models has been computed. It has been found that SPC/E and TIP5P do not correctly describe the phase diagram of water. However, TIP4P provides a qualitatively correct description of the phase diagram of water. A slight modification of the parameters of the TIP4P model yields a new model, denoted as TIP4P/ice, which reproduces the experimental melting point of water and provides an excellent description of the densities of all ice phases.

Vapor-liquid interfacial properties of fully flexible Lennard-Jones chains

We consider the computation of the interfacial properties of molecular chains from direct simulation of the vapor-liquid interface. The molecules are modeled as fully flexible chains formed from tangentially bonded monomers with truncated Lennard-Jones interactions. Four different model systems comprising of 4, 8, 12, and 16 monomers per molecule are considered. The simulations are performed in the canonical ensemble, and the vapor-liquid interfacial tension is evaluated using the test area and the wandering interface methods. In addition to the surface tension, we also obtain density profiles, coexistence densities, critical temperature and density, and interfacial thickness as functions of temperature, paying particular attention to the effect of the chain length on these properties. According to our results, the main effect of increasing the chain length (at fixed temperature) is to sharpen the vapor-liquid interface and to increase the width of the biphasic coexistence region. As a result, the interfacial thickness decreases and the surface tension increases as the molecular chains get longer. The interfacial thickness and surface tension appear to exhibit an asymptotic limiting behavior for long chains. A similar behavior is also observed for the coexistence densities and critical properties. Our simulation results indicate that the asymptotic regime is reached for Lennard-Jones chains formed from eight monomer segments. We also include a preliminary study on the effect of the cutoff distance on the interfacial properties. Our results indicate that all of the properties exhibit a dependence with the distance at which the interactions are truncated, though the relative effect varies from one property to the other. The interfacial thickness and, more particularly, the interfacial tension are found to be strongly dependent on the particular choice of cutoff, whereas the density profiles and coexistence densities are, in general, less sensitive to the truncation.

Nano-dewetting: Interplay between van der Waals- and short-ranged interactions

The wetting behavior of a polymer liquid on a coated substrate is investigated via numerical self-consistent field calculations and experiments. The polymer does not wet the coated substrate, but the substrate might stabilize a mesoscopic film. Experiments observe a stable mesoscopic film of nanometer thickness in coexistence with macroscopic drops at high temperatures. Upon cooling, this mesoscopic polymer film breaks up into droplets (nano-dewetting). Our self-consistent field calculations suggest that the stability of the mesoscopic film is determined by a subtle interplay between van der Waals forces and short-ranged forces due to the distortion of the profile in the vicinity of the substrate.

The range of meta stability of ice-water melting for two simple models of water

A number of crystal structures of water have been ‘superheated’ in Monte Carlo simulations. Two well-known models for water were considered; namely the TIP4P model and the SPC/E model. By comparing the fluid–solid coexistence temperature to the temperature at which the solid becomes mechanically unstable and melts it is possible to determine the typical range of temperatures over which it is possible to superheat the ice phases in conventional simulation studies. It is found that the ice phases can be superheated to approximately 90 K beyond the fluid–solid coexistence temperature. Beyond this limit they spontaneously melt. This limit appears to depend weakly both on the type of ice phase considered and on the chosen model. Obviously only rigorous free energy calculations can determine the equilibrium fluid–solid coexistence of a model. However, a ‘rule of thumb’ is that, by subtracting 90 K from the mechanical stability limit of the ice phase one is provided with a first guess as to the equilibrium fluid–solid coexistence temperature.

Combinatorial entropy and phase diagram of partially ordered ice phases

A close analytical estimate for the combinatorial entropy of partially ordered ice phases is presented. The expression obtained is very general, as it can be used for any ice phase obeying the Bernal-Fowler rules. The only input required is a number of crystallographic parameters, and the experimentally observed proton site occupancies. For fully disordered phases such as hexagonal ice, it recovers the result deduced by Pauling, while for fully ordered ice it is found to vanish. Although the space groups determined for ice I, VI, and VII require random proton site occupancies, it is found that such random allocation of protons does not necessarily imply random orientational disorder. The theoretical estimate for the combinatorial entropy is employed together with free energy calculations in order to obtain the phase diagram of ice from 0 to 10 GPa. Overall qualitative agreement with experiment is found for the TIP4P model of water. An accurate estimate of the combinatorial entropy is found to play an important role in determining the stability of partially ordered ice phases, such as ice III and ice V.

Extending Wertheim’s perturbation theory to the solid phase: The freezing of the pearl-necklace model

An extension of Wertheim’s first order perturbation theory [M. S. Wertheim, J. Chem. Phys. 87, 7323 (1987)] for chain molecules in the solid phase is proposed. In particular the solid phase of a model of freely jointed chains of tangent hard spheres has been considered. The equation of state and free energies from theory are compared with simulation results obtained previously by Malanoski and Monson [A. P. Malanoski and P. A. Monson, J. Chem. Phys. 107, 6899 (1997)] for chains with up to eight monomer units. The agreement between theory and simulation is good for the densities close to the melting, and deteriorates for densities close to close-packing. We also evaluated theoretically the fluid-solid equilibrium of the pearl-necklace model and compare with simulation results. The theory reproduces reasonably well the trends observed in the simulation results.

Coarse-grained models for fluids and their mixtures: Comparison of Monte Carlo studies of their phase behavior with perturbation theory and experiment.

The prediction of the equation of state and the phase behavior of simple fluids (noble gases, carbon dioxide, benzene, methane, and short alkane chains) and their mixtures by Monte Carlo computer simulation and analytic approximations based on thermodynamic perturbation theory is discussed. Molecules are described by coarse grained models, where either the whole molecule (carbon dioxide, benzene, and methane) or a group of a few successive CH(2) groups (in the case of alkanes) are lumped into an effective point particle. Interactions among these point particles are fitted by Lennard-Jones (LJ) potentials such that the vapor-liquid critical point of the fluid is reproduced in agreement with experiment; in the case of quadrupolar molecules a quadrupole-quadrupole interaction is included. These models are shown to provide a satisfactory description of the liquid-vapor phase diagram of these pure fluids. Investigations of mixtures, using the Lorentz-Berthelot (LB) combining rule, also produce satisfactory results if compared with experiment, while in some previous attempts (in which polar solvents were modeled without explicitly taking into account quadrupolar interaction), strong violations of the LB rules were required. For this reason, the present investigation is a step towards predictive modeling of polar mixtures at low computational cost. In many cases Monte Carlo simulations of such models (employing the grand-canonical ensemble together with reweighting techniques, successive umbrella sampling, and finite size scaling) yield accurate results in very good agreement with experimental data. Simulation results are quantitatively compared to an analytical approximation for the equation of state of the same model, which is computationally much more efficient, and some systematic discrepancies are discussed. These very simple coarse-grained models of small molecules developed here should be useful, e.g., for simulations of polymer solutions with such molecules as solvent.