Ideal Gas Contribution Research Articles

Deriving phase field crystal theory from dynamical density functional theory: Consequences of the approximations.

Phase field crystal (PFC) theory is extensively used for modeling the phase behavior, structure, thermodynamics, and other related properties of solids. PFC theory can be derived from dynamical density functional theory (DDFT) via a sequence of approximations. Here, we carefully identify all of these approximations and explain the consequences of each. One approximation that is made in standard derivations is to neglect a term of form ∇·[n∇Ln], where n is the scaled density profile and L is a linear operator. We show that this term makes a significant contribution to the stability of the crystal, and that dropping this term from the theory forces another approximation, that of replacing the logarithmic term from the ideal gas contribution to the free energy with its truncated Taylor expansion, to yield a polynomial in n. However, the consequences of doing this are (i) the presence of an additional spinodal in the phase diagram, so the liquid is predicted first to freeze and then to melt again as the density is increased; and (ii) other periodic structures, such as stripes, are erroneously predicted to be thermodynamic equilibrium structures. In general, L consists of a nonlocal convolution involving the pair direct correlation function. A second approximation sometimes made in deriving PFC theory is to replace L with a gradient expansion involving derivatives. We show that this leads to the possibility of the density going to zero, with its logarithm going to -∞ while being balanced by the fourth derivative of the density going to +∞. This subtle singularity leads to solutions failing to exist above a certain value of the average density. We illustrate all of these conclusions with results for a particularly simple model two-dimensional fluid, the generalized exponential model of index 4 (GEM-4), chosen because a DDFT is known to be accurate for this model. The consequences of the subsequent PFC approximations can then be examined. These include the phase diagram being both qualitatively incorrect, in that it has a stripe phase, and quantitatively incorrect (by orders of magnitude) regarding the properties of the crystal phase. Thus, although PFC models are very successful as phenomenological models of crystallization, we find it impossible to derive the PFC as a theory for the (scaled) density distribution when starting from an accurate DDFT, without introducing spurious artifacts. However, we find that making a simple one-mode approximation for the logarithm of the density distribution lnρ(x) rather than for ρ(x) is surprisingly accurate. This approach gives a tantalizing hint that accurate PFC-type theories may instead be derived as theories for the field lnρ(x), rather than for the density profile itself.

Nonzero Ideal Gas Contribution to the Surface Tension of Water.

Surface tension, the tendency of fluid interfaces to behave elastically and minimize their surface, is routinely calculated as the difference between the lateral and normal components of the pressure or, invoking isotropy in momentum space, of the virial tensor. Here we show that the anisotropy of the kinetic energy tensor close to a liquid-vapor interface can be responsible for a large part of its surface tension (about 15% for water, independent from temperature).

Uncertainty assessment of equations of state with application to an organic Rankine cycle

ABSTRACTEvaluations of equations of state (EoS) should include uncertainty. This study presents a generic method to analyse EoS from a detailed uncertainty analysis of the mathematical form and the data used to obtain EoS parameter values. The method is illustrated by comparison of Soave–Redlich–Kwong (SRK) cubic EoS with perturbed-chain statistical associating fluid theory (PC-SAFT) EoS for an organic Rankine cycle (ORC) for heat recovery to power from the exhaust gas of a marine diesel engine using cyclopentane as working fluid. Uncertainties of the EoS input parameters including their corresponding correlation structure, are quantified from experimental measurements using a bootstrap method. Variance-based sensitivity analysis is used to compare the uncertainties from the departure function and the ideal-gas contribution. A Monte Carlo procedure propagates fluid parameter input uncertainty onto the model outputs. Uncertainties in the departure function (SRK or PC-SAFT EoS) dominate the total uncertainties of the ORC model output. For this application and working fluid, SRK EoS has less predictive uncertainty in the process model output than does PC-SAFT EoS, though it cannot be determined if this is due to differences in the data for parameter estimation or in the mathematical form of the EoS or both.

Different ways of looking at the force between two nanocrystals.

The potential of mean force (PMF) between two nanocrystals (NCs) represents an effective interaction potential that is essential when explaining the assembly of NCs to superstructures. For a given temperature, the PMF is obtained best from molecular dynamics simulations. Based on a density functional approach, this study proposes three methods of predicting the PMF for any given temperature based on a single molecular dynamics simulation for one temperature. The three methods construct the PMF by considering the ligands as an ideal gas, as hard-sphere chains, or as Lennard-Jones interaction sites. To apply this methodology, the density of the interaction centers must be extracted from the simulation data. For the ideal gas model, a straightforward sampling procedure with a fixed lattice in space leads to free energies that are too large in order to consistently explain the simulation data for different temperatures. Naive sampling does not account for the small momenta added to the NCs when coupled to a thermostat. A method is proposed that corrects for the unphysical steps during the simulation. The ideal gas contribution computed for the corrected density is significantly smaller than the one obtained from naive sampling and can thus explain the temperature dependence of the PMF correctly. For the hard-sphere chain model, where a weighted density is used, the correction of the particle density is not essential. However, the PMF calculated based on the corrected density confirms our approach. All three models predict PMF curves in very good agreement with simulation results, but they differ in the number of input parameters and the computational effort. Based on the modeling results, we predict the existence of an additional attractive force at small distances of the NCs - a depletion force.

Ideal gas contribution to the isobaric heat capacity of refrigerants: Poling et al.’s polynomial correlation vs DIPPR data

The ideal gas contribution to the isobaric heat capacity of fluids is a temperature dependent property which is commonly modelled as a polynomial expression. In this work, the performance and accuracy of the polynomial correlation proposed by Poling et al. in their well-known book is checked. To this end, the data accepted in the DIPPR database for 58 refrigerants were used. The mean average percentage deviations (MAPDs) obtained in the temperature range considered by Poling et al., usually up to 1000K, are greater than 1.5% only for six refrigerants. We extended our study to the temperature range in which accepted DIPPR data are available (usually up to 1500K), finding that only for four refrigerants can the Poling et al. correlation be used in this extended range. New coefficients for the correlation are given for the 58 refrigerants studied which reproduce the accepted DIPPR data. The new MAPD values are then below 1% for 49 refrigerants.

The influence of electron degeneracy on the contribution of bound states to the non-ideal hydrogen plasma EOS

An equation of state for a weakly non-ideal hydrogen plasma was developed to account for the influence of degenerate electrons on the contribution of bound states. Asymptotic expressions for the contribution were derived and compared. In this work, the reduced model EOS includes the ideal gas contribution with degenerate electrons and relativistic corrections, bound states contribution and the Coulomb interaction in the Debye–Hückel approximation. The influence of the electron degeneracy on the adiabatic exponent and the total pressure is shown.

Equations of State for Mixtures of R-32, R-125, R-134a, R-143a, and R-152a

Mixture models explicit in Helmholtz energy have been developed to calculate the thermodynamic properties of refrigerant mixtures containing R-32, R-125, R-134a, R143a, and R-152a. The Helmholtz energy of the mixture is the sum of the ideal gas contribution, the compressibility (or real fluid) contribution, and the contribution from mixing. The independent variables are the density, temperature, and composition. The model may be used to calculate the thermodynamic properties of mixtures, including dew and bubble point properties, within the experimental uncertainties of the available measured properties. It incorporates the most accurate equations of state available for each pure fluid. The estimated uncertainties of calculated properties are 0.1% in density and 0.5% in heat capacities and in the speed of sound. Calculated bubble point pressures have typical uncertainties of 0.5%.

Structure of Polyelectrolyte Solutions at a Charged Surface

The structure of polyelectrolyte solutions at a charged surface is studied using density functional theory and Monte Carlo simulations. The polymer molecules are modeled as freely jointed chains of charged hard spheres, the counterions and co-ions as charged hard spheres, and the surface is a planar, impenetrable hard wall with a uniform surface charge density. The density functional theory treats the ideal gas contribution exactly, uses a weighted density approximation for the hard sphere contribution, and a generalized van der Waals approximation for the electrostatic contribution. At a fixed surface charge density, with increasing concentration of the polyion, the simulations show layering as well as charge inversion phenomena resulting from an interplay between excluded volume and electrostatic interactions. The integrated surface excess is a monotonically decreasing function of polymer concentration at high surface charge densities, but a nonmonotonic function of polymer concentration at lower surfac...

Density Functional Theory of Square-well Chain Mixtures Near Solid Surface

A density functional theory is developed for square-well chain mixtures near a hard wall or a square-well wall. The Helmholtz free energy functional is divided into two parts, an ideal-gas contribution and an excess contribution. The later is constructed based on weighting density approximation and coarse graining approximation. For simplicity, a Heaviside function is used as the weighting function. To calculate the density profiles of beads, a direct iterative algorithm combined with a relaxation factor is used. A single chain simulation in an effective field is needed in each iterative step. The theoretical results are in good agreement with Monte Carlo simulation.

Theoretical study for partial molar volume of amino acids in aqueous solution: Implication of ideal fluctuation volume

A Kirkwood–Buff equation for the partial molar volumes of polyatomic molecules in solutions is derived based on the reference interaction site model (RISM) theory of molecular liquids. The partial molar volume of the twenty amino acids in aqueous solution at infinite dilution are calculated using the equation, and the results are discussed in comparison with the experimental data. The results indicate that ionizations of the C- and N-terminus groups give negative contributions to the volume ranging from −3.2 cm3/mol to −9.7 cm3/mol depending on the amino acid. Ionization of the dissociable residues also give negative contribution which ranges from −3.0 cm3/mol to −6.0 cm3/mol. On the other hand, contribution of the fractional charges on atoms to the volume is not necessarily negative, but rather slightly positive with few exceptions. It is clarified that contribution from an atom group to the volume is largely dependent on the situation where the group is placed. Therefore, it is concluded that the conventional way of determining the partial molar volume from group contributions is not reliable. The theoretical results for the partial molar volume exhibit a systematic deviation from corresponding experimental data, which increases nearly proportionally with increasing temperature and with the number of atoms in the amino acids. In order to account for the deviation, a concept of the “ideal fluctuation volume” is proposed, which is the ideal gas contribution to the volume, originating from the intramolecular fluctuation of solute.

Thermodynamic Properties of Air and Mixtures of Nitrogen, Argon, and Oxygen From 60 to 2000 K at Pressures to 2000 MPa

A thermodynamic property formulation for standard dry air based upon available experimental p–ρ–T, heat capacity, speed of sound, and vapor–liquid equilibrium data is presented. This formulation is valid for liquid, vapor, and supercritical air at temperatures from the solidification point on the bubble-point curve (59.75 K) to 2000 K at pressures up to 2000 MPa. In the absence of reliable experimental data for air above 873 K and 70 MPa, air properties were predicted from nitrogen data in this region. These values were included in the determination of the formulation to extend the range of validity. Experimental shock tube measurements on air give an indication of the extrapolation behavior of the equation of state up to temperatures and pressures of 5000 K and 28 GPa. The available measurements of thermodynamic properties of air are summarized and analyzed. Separate ancillary equations for the calculation of dew and bubble-point pressures and densities of air are presented. In the range from the solidification point to 873 K at pressures to 70 MPa, the estimated uncertainty of density values calculated with the equation of state is 0.1%. The estimated uncertainty of calculated speed of sound values is 0.2% and that for calculated heat capacities is 1%. At temperatures above 873 K and 70 MPa, the estimated uncertainty of calculated density values is 0.5% increasing to 1.0% at 2000 K and 2000 MPa. In addition to the equation of state for standard air, a mixture model explicit in Helmholtz energy has been developed which is capable of calculating the thermodynamic properties of mixtures containing nitrogen, argon, and oxygen. This model is valid for temperatures from the solidification point on the bubble-point curve to 1000 K at pressures up to 100 MPa over all compositions. The Helmholtz energy of the mixture is the sum of the ideal gas contribution, the real gas contribution, and the contribution from mixing. The contribution from mixing is given by a single generalized equation which is applied to all mixtures used in this work. The independent variables are the reduced density and reduced temperature. The model may be used to calculate the thermodynamic properties of mixtures at various compositions including dew and bubble-point properties and critical points. It incorporates the most accurate published equation of state for each pure fluid. The mixture model may be used to calculate the properties of mixtures generally within the experimental accuracies of the available measured properties. The estimated uncertainty of calculated properties is 0.1% in density, 0.2% in the speed of sound, and 1% in heat capacities. Calculated dew and bubble-point pressures are generally accurate to within 1%.

A Generalized Model for the Thermodynamic Properties of Mixtures

A mixture model explicit in Helmholtz energy has been developed which is capable of predicting thermodynamic properties of mixtures containing nitrogen, argon, oxygen, carbon dioxide, methane, ethane, propane, n-butane, i-butane, R-32, R-125, R-134a, and R-152a within the estimated accuracy of available experimental data. The Helmholtz energy of the mixture is the sum of the ideal gas contribution, the compressibility (or real gas) contribution, and the contribution from mixing. The contribution from mixing is given by a single generalized equation which is applied to all mixtures studied in this work. The independent variables are the density, temperature, and composition. The model may be used to calculate the thermodynamic properties of mixtures at various compositions including dew and bubble point properties and critical points. It incorporates accurate published equations of state for each pure fluid. The estimated accuracy of calculated properties is ±0.2% in density, ±0.1 % in the speed of sound at pressures below 10 MPa, ±0.5% in the speed of sound for pressures above 10 MPa, and ±1% in heat capacities. In the region from 250 to 350 K at pressures up to 30 MPa, calculated densities are within ±0.1 % for most gaseous phase mixtures. For binary mixtures where the critical point temperatures of the pure fluid constituents are within 100 K of each other, calculated bubble point pressures are generally accurate to within ±1 to 2%. For mixtures with critical points further apart, calculated bubble point pressures are generally accurate to within ±5 to 10%.

Thermodynamic Properties of Mixtures of R-32, R-125, R-134a, and R-152a

A mixture model explicit in Helmholtz energy has been developed that is capable of predicting thermodynamic properties of refrigerant mixtures containing R-32, R-125, R-134a, and R-152a. The Helmholtz energy of the mixture is the sum of the ideal gas contribution, the compressibility (or real gas) contribution, and the contribution from mixing. The contribution from mixing is given by a single equation that is applied to all mixtures used in this work. The independent variables are the density, temperature, and composition. The model may be used to calculate thermodynamic properties of mixtures, including dew and bubble point properties and critical points, generally within the experimental uncertainties of the available measured properties. It incorporates the most accurate published equation of state for each pure fluid. The estimated uncertainties of calculated properties are ±0.25% in density, ±0.5% in the speed of sound, and ±1% in heat capacities. Calculated bubble point pressures are generally accurate to within ±1%.

Classification of diagrams for a plasma equation-of-state

Utilizing the Montroll-Ward approach to quantum statistics, generalized to many components, we seek an equation of state for a high-temperature low-density plasma. We propose a classification of diagrams analogous to that used by Meeron and by Friedman in classical statistical mechanics. In the resulting expansion, the leading term is the ideal gas contribution plus the lowest-order exchange contribution plus the ring contribution, the latter representing the effect upon the pressure of collective motions of a completely ionized plasma. The next term appears to represent a contribution due to modifications of free particle motion due to the interaction of the single particles with the rest of the plasma. The third term represents the contribution of two-particle states, both of positive energy between all particle pairs, and negative energy (bound states) between pairs of opposite charge sign. The third term also contains parts representing modifications of the motion of the pair of particles due to interaction of the individual particles with the rest of the plasma. Higher terms represent, successively, contributions of three-, four-, etc., particle states, again including modifications of the single-particle motions. The classification automatically eliminates the ultraviolet catastrophe which ordinarily arises in the treatment of Coulomb force bound states in statistical mechanics, since, in effect, it uses a screened Coulomb potential instead of the ordinary potential. In addition, the short range divergence, which occurs in classical theory because of the `fall' of the electron to the nucleus, does not arise, being prevented by the uncertainty principle.