Constant Volume Excess Heat Capacity Research Articles

Thermodynamic properties of diamond and wurtzite model fluids from computer simulation and thermodynamic perturbation theory

Monte Carlo NVT simulations have been performed to obtain the thermodynamic and structural properties and perturbation coefficients up to third order in the inverse temperature expansion of the Helmholtz free energy of fluids with potential models proposed in the literature for diamond and wurtzite lattices. These data are used to analyze performance of a coupling parameter series expansion (CPSE). The main findings are summarized as follows, (1) The CPSE provides accurate predictions of the first three coefficient in the inverse temperature expansion of Helmholtz free energy for the potential models considered and the thermodynamic properties of these fluids are predicted more accurately when the CPSE is truncated at second or third order. (2) The Barker–Henderson (BH) recipe is appropriate for determining the effective hard sphere diameter for strongly repulsive potential cores, but its performance worsens with increasing the softness of the potential core. (3) For some thermodynamic properties the first-order CPSE works better for the diamond potential, whose tail is dominated by repulsive interactions, than for the potential, whose tail is dominated by attractive interactions. However, the first-order CPSE provides unsatisfactory results for the excess internal energy and constant-volume excess heat capacity for the two potential models.

A comprehensive comparison between thermodynamic perturbation theory and first-order mean spherical approximation: Based on discrete potentials with hard core

Using the TL (Tang and Lu, 1993) method, Ornstein-Zernike integral equation is solved perturbatively under the mean spherical approximation (MSA) for fluid with potential consisting of a hard sphere plus square-well plus square-shoulder (HS+SW+SS) to obtain first-order analytic expressions of radial distribution function (RDF), second-order direct correlation function, and semi-analytic expressions for common thermodynamic properties. A comprehensive comparison between the first-order MSA and high temperature series expansion (HTSE) to third-, fifth- and seventh-order is performed over a wide parameter range for both a HS+SW and the HS+SW+SS model fluids by using corresponding “exact” Monte Carlo results as a reference; although the HTSE is carried out up to seventh-order, and not to the first order as the first-order MSA the comparison is considered fair from a calculation complexity perspective. It is found that the performance of the first-order MSA is dramatically model-dependent: as target potentials go from the HS+SW to the HS+SW+SS, (i) there is a dramatic dropping of performance of the first-order MSA expressions in calculating the thermodynamic properties, especially both the excess internal energy and constant volume excess heat capacity of the HS+SW+SS model cannot be predicted even qualitatively correctly. (ii) One tendency is noticed that the first-order MSA gets more reliable with increasing temperatures in dealing with the pressure, excess Helmholtz free energy, excess enthalpy and excess chemical potential. (iii) Concerning the RDF, the first-order MSA is not as disappointing as it displays in the cases of thermodynamics. (iv) In the case of the HS+SW model, the first-order MSA solution is shown to be quantitatively correct in calculating the pressure and excess chemical potential even if the reduced temperatures are as low as 0.8. On the other hand, the seventh-order HTSE is less model-dependent; in most cases of the HS+SW and the HS+SW+SS models, the seventh-order HTSE improves the fifth- and third-order HTSE in both thermodynamic properties and RDF, and the improvements are very demonstrable in both the excess internal energy and constant volume excess heat capacity; for very limited cases, the seventh-order HTSE improves the fifth-order HTSE only within lower density domain and even shows a bit of inadaptation over higher density domain.

Coupling parameter series expansion for fluid with square-well plus repulsive-square-barrier potential

Monte Carlo simulations in the canonical ensemble are performed for fluid with potential consisting of a square-well plus a square-barrier to obtain thermodynamic properties such as pressure, excess energy, constant volume excess heat capacity, and excess chemical potential, and structural property such as radial distribution function. The simulations cover a wide density range for the fluid phase, several temperatures, and different combinations of the parameters defining the potential. These simulation data have been used to test performances of a coupling parameter series expansion (CPSE) recently proposed by one of the authors [S. Zhou, Phys. Rev. E 74, 031119 (2006)], and a traditional 2nd-order high temperature series expansion (HTSE) based on a macroscopic compressibility approximation (MAC) used with confidence since its introduction in 1967. It is found that (i) the MCA-based 2nd-order HTSE unexpectedly and depressingly fails for most situations investigated, and the present simulation results can serve well as strict criteria for testing liquid state theories. (ii) The CPSE perturbation scheme is shown to be capable of predicting very accurately most of the thermodynamic properties simulated, but the most appropriate level of truncating the CPSE differs and depends on the range of the potential to be calculated; in particular, the shorter the potential range is, the higher the most appropriate truncating level can be, and along with rising of the potential range the performance of the CPSE perturbation scheme will decrease at higher truncating level. (iii) The CPSE perturbation scheme can calculate satisfactorily bulk fluid rdf, and such calculations can be done for all fluid states of the whole phase diagram. (iv) The CPSE is a convergent series at higher temperatures, but show attribute of asymptotic series at lower temperatures, and as a result, the surest asymptotic value occurs at lower-order truncation.

Convergence and low temperature adaptability analysis of the high temperature series expansion of the free energy

By appealing to the coupling parameter series expansion to calculate the first seven perturbation coefficients of the high temperature series expansion (HTSE) of the free energy, analysis of convergence and low temperature adaptability of the HTSE in calculating fluid thermodynamic properties is performed for the first time; the fluid thermodynamic properties considered include critical parameters, vapor-liquid coexistence curve, thermodynamic characteristic functions, chemical potential, pressure, and constant volume excess heat capacity. To proceed with the analysis, a well known square well model is used as sample; the well widths considered range over a wide interval, and the relevant temperatures amenable to simulation calculations (used as "exact" results to analyze the HTSE) can be both very high and very low. The main discoveries reached are summarized as follows: (1) The HTSE usually converges at the 4th-order truncation, but with decrease of the temperature considered, the lowest truncation order, which makes the HTSE to converge, tends to rise. As a conservative estimate, it is considered that the HTSE always converges for reduced temperature T* higher than 0.25, whereas for T* < 0.25 there appear signs indicating that the HTSE may diverge from the 7th-order truncation. (2) Within the temperature interval with T* ≥ 0.5, the HTSE converges approximately to the correct solution, and the HTSE can be reliably used to calculate the fluid thermodynamic properties, and within this temperature interval, the 4th-order truncation is enough; whereas for T* < 0.5, such as within the temperature interval with 0.275 ≤ T* ≤ 0.355, although the HTSE does converge, it does not converge to the correct solution, and the deviations between the HTSE calculations and MC simulations become an ever-prominent issue with the rising of the density, and the slopes of the thermodynamic properties over density are not satisfactorily represented. As a result, the HTSE is not suited for calculations for temperature interval T* < 0.5.

Non hard sphere thermodynamic perturbation theory over a wide range of temperatures

A recently proposed non hard sphere (HS) coupling parameter expansion (CPE)thermodynamic perturbation theory (TPT) is evaluated for its performance over a widerange of temperatures, particularly in tackling a so-called low temperature problem,i.e. the traditional high temperature series expansion TPT performance becomesprogressively less satisfactory as the temperature drops, and finally becomes qualitativelyincorrect. Accordingly, we have performed Monte Carlo simulations in the canonicalensemble for a honeycomb model potential, and the pressure, excess internalenergy, excess Helmholtz free energy, excess chemical potential, and excess enthalpyhave been obtained over wide density and temperature ranges for the fluid phase.These new simulation data, together with published simulation data for an HS + square well fluid at very low temperatures, have been used to test the performanceof the non HS CPE third-order TPT. A general scheme for assimilating partof the tail term of the potential function considered into a reference system isproposed, which ensures running normality of the non HS perturbation codeat subcritical temperatures and smooth transition into a commonly used HSperturbation scheme when the temperatures considered rise infinitely. It is found thatthe non HS CPE third-order TPT is very satisfactory or even very accurate forthese extremely low temperatures for which traditional Zwanzig type TPT isqualitatively incorrect and even an HS CPE third-order TPT also seriously fails insome cases. Among the thermodynamic quantities considered the most difficultone to predict theoretically is the constant volume excess heat capacity, and forthis quantity the non HS CPE third-order TPT is also obviously superior to itscompetitors.

Low temperature behavior of thermodynamic perturbation theory

Monte Carlo simulations have been carried out for a hard sphere square well model fluid with well widths of lambda = 1.01, 1.02 and 1.04 to obtain thermodynamic properties, such as pressure, excess Helmholtz free energy and internal energy, constant volume excess heat capacity, excess chemical potential, and excess enthalpy. The extremely narrow well widths considered allowed us to perform simulations at extremely low temperatures without reaching the liquid-vapour transition, which is metastable for these values of lambda. These simulation data have been used to explore the low temperature behavior of two thermodynamic perturbation theories (TPT), namely a high temperature series expansion truncated at second order within the local compressibility approximation (thereafter denoted as 2nd-order LCA-TPT) and a recently proposed coupling parameter series expansion truncated at 3rd-, 4th- and 5th-order (thereafter denoted as 3rd-order, 4th-order and 5th-order TPT, respectively). It has been found that the 2nd-order LCA-TPT is qualitatively incorrect in most of the cases analyzed, whereas the 3rd-order TPT is quantitatively correct in most of them. With increasing the well width, and consequently the temperatures considered, the 3rd-order TPT quickly becomes more accurate. Among the six thermodynamic quantities analyzed, the one most difficult to predict accurately is the constant volume excess heat capacity, for which the TPT based on the coupling parameter expansion provides satisfactory results only in the case of lambda = 1.04. For the cases studied, the performance of the 3rd-order, 4th-order, and 5th-order TPT are essentially equal; the change of the order of the truncation sometimes results in an improvement of some thermodynamic quantities within certain regions of the thermodynamic surface of states, while worsening in others.

Thermodynamic properties of double square-well fluids: Computer simulations and theory

Computer simulations have been performed to obtain the thermodynamic properties of fluids with double square-well potentials, that is, potentials consisting of two adjacent square wells with different depths. The compressibility factor, excess energy, chemical potential, constant-volume excess heat capacity, and other derived properties have been obtained. These data have been used to test the performance of several perturbation theories for predicting the thermodynamic properties of this kind of fluids. Good agreement is found on the whole between theory and simulation at supercritical temperatures. The possible presence of anomalous behavior at high densities in the fluids considered has been also analyzed in light of the same theories, with the result that in general, they do not predict such anomalous behavior, with the possible exception of a Monte Carlo-based perturbation theory for relatively large potential widths at high densities and very low temperatures.

Third-order thermodynamic perturbation theory for effective potentials that model complex fluids

We have performed Monte Carlo simulations to obtain the thermodynamic properties of fluids with two kinds of hard-core plus attractive-tail or oscillatory potentials. One of them is the square-well potential with small well width. The other is a model potential with oscillatory and decaying tail. Both model potentials are suitable for modeling the effective potential arising in complex fluids and fluid mixtures with extremely-large-size asymmetry, as is the case of the solvent-induced depletion interactions in colloidal dispersions. For the former potential, the compressibility factor, the excess energy, the constant-volume excess heat capacity, and the chemical potential have been obtained. For the second model potential only the first two of these quantities have been obtained. The simulations cover the whole density range for the fluid phase and several temperatures. These simulation data have been used to test the performance of a third-order thermodynamic perturbation theory (TPT) recently developed by one of us [S. Zhou, Phys. Rev. E 74, 031119 (2006)] as compared with the well-known second-order TPT based on the macroscopic compressibility approximation due to Barker and Henderson. It is found that the first of these theories provides much better accuracy than the second one for all thermodynamic properties analyzed for the two effective potential models.

A new coordination number model and heat capacity of square-well fluids of variable width

A new coordination number model for square-well fluid is proposed in this work. The model modifies the theory previously developed for the coordination number of square-well fluids derived on the basis of the generalized van der Waals theory by Largo and Solana. In the present work, we have calculated the co-ordination number of a square-well fluid on the basis of high temperature expansion (HTE) approach along with the modified formula of the maximum reduced density and a novel approach to derive the coordination number of hard sphere fluids. We have also performed the theoretical predictions of constant-volume excess heat capacity of the square-well fluids. It is shown that the theoretical predictions of the constant-volume excess heat capacity determined by second-order perturbation term are in much better agreement with simulation data than to the values of C V E / N K obtained by other workers due to the better description of the second-order term.

Thermodynamic properties of square-well fluids from a new coordination number model

A new coordination number model is proposed for square-well fluids at λ = 1.5. The model modifies the theory previously developed for the coordination number of square-well fluids derived on the basis of the generalized Van der Waals theory by Largo and Solana. The present work proposes a new formula for the maximum reduced density and a new method to solve the integral corresponding to coordination number of hard sphere fluids. Thermodynamic quantities such as pair correlation function at contact, excess internal energy and constant volume excess heat capacity are determined employing the model. Good agreement is achieved between theoretical results and Monte Carlo (M.C.) data. As expected, the results of the excess heat capacity are poor at low temperatures.

Heat capacity of square-well fluids of variable width

We have obtained by Monte Carlo NVT simulations the constant-volume excess heat capacity of square-well fluids for several temperatures, densities and potential widths. Heat capacity is a thermodynamic property much more sensitive to the accuracy of a theory than other thermodynamic quantities, such as the compressibility factor. This is illustrated by comparing the reported simulation data for the heat capacity with the theoretical predictions given by the Barker-Henderson perturbation theory as well as with those given by a non-perturbative theoretical model based on Baxter's solution of the Percus-Yevick integral equation for sticky hard spheres. Both theories give accurate predictions for the equation of state. By contrast, it is found that the Barker-Henderson theory strongly underestimates the excess heat capacity for low to moderate temperatures, whereas a much better agreement between theory and simulation is achieved with the non-perturbative theoretical model, particularly for small well widths, although the accuracy of the latter worsens for high densities and low temperatures, as the well width increases.