Article

# Critical Isotherms from Virial Series Using Asymptotically Consistent Approximants

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## Abstract

We consider the low-density equation of state of a fluid along its critical isotherm. An asymptotically consistent approximant is formed having the correct leading-order scaling behavior near the vapor-liquid critical point, while retaining the correct low-density behavior as expressed by the virial equation of state. The formulation is demonstrated for the Lennard-Jones fluid, and models for helium, water, and n-alkanes. The ability of the approximant to augment virial-series predictions of critical properties is explored, both in conjunction with and in the absence of critical-property data obtained by other means. Given estimates of the critical point from molecular simulation or experiment, the approximant can refine the critical pressure or density by ensuring that the critical isotherm remains well-behaved from low density to the critical region. Alternatively, when applied in the absence of other data, the approximant remedies a consistent underestimation of the critical density when computed from the virial series alone. © 2014 American Institute of Chemical Engineers AIChE J, 2014

## Supplementary resources (3)

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... Approximants have been used to examine a single path of approach for lattice models (viz., along the critical isochore) 7,14-16 and for molecular model fluids (along the critical isotherm). 17 Here, we are interested in an analytic continuation of the virial series that incorporates the first three scaling paths given in Eq. (2), so that we may obtain an equation of state valid at low density, the critical region, and intermediate regions. Care is required to ensure that singularities enforced at the critical point do not introduce anomalies away from the critical region. ...
... Although several choices lead to a convergent description of the SW fluid, we use f (θ) = θ in what follows, as this leads to a desirable scaling property (discussed below). The choice of the r-dependence in the auxiliary function given in Eq. (4) allows the approximant to reduce to a critical isotherm approximant when T = T c , similar to the one given in Ref. 17, such that P = P c − A(ρ)(1 − ρ/ρ c ) δ . The approximant given by Eq. (4) at T = T c differs from the one in Ref. 17 in that it relies on knowledge of B 0 , and is not restricted to ρ < ρ c . ...
... The choice of the r-dependence in the auxiliary function given in Eq. (4) allows the approximant to reduce to a critical isotherm approximant when T = T c , similar to the one given in Ref. 17, such that P = P c − A(ρ)(1 − ρ/ρ c ) δ . The approximant given by Eq. (4) at T = T c differs from the one in Ref. 17 in that it relies on knowledge of B 0 , and is not restricted to ρ < ρ c . ...
Conference Paper
The virial equation of state (VEOS) is a power series in density, with coefficients that can be determined from molecular considerations. Its rigorous connection to statistical mechanics makes it an appealing choice for many purposes, but its utility is limited to conditions where the series converges. Singular behavior in the equation of state, such as encountered upon approach to the critical point, restricts its range of application. In previous work, we took steps to address this problem by forming an approximant that explicitly incorporates this singular behavior while remaining fully consistent with the VEOS at low density. This treatment was shown to provide several new capabilities, including improved predictions of critical properties from the virial coefficients, refinement of critical properties given by simulation/experimental measurements, and an estimate of a critical amplitude for several fluids. Our previous work was restricted to the critical isotherm. In the present work, we enhance the approximant by including temperature dependence, allowing us to incorporate scaling along the critical isochore. The approximant is designed to enforce the correct singular behavior at the vapor-liquid critical point, while still retaining the correct temperature-dependent low-density behavior as expressed by the VEOS. The efficacy of the proposed equation of state is demonstrated for the square-well fluid, chosen for this initial study because the temperature dependence of its virial coefficients can be expressed analytically.
... The expression we evaluate for the second virial coefficient is 10 (1) , Z (2) ). ...
... We define F (2) with an argument p which may be but is not necessarily equal to P. Also, we have introduced here the dimensionless temperature-dependent parameter k h , which relates to the strength of the harmonic "spring constant" between adjacent orientation beads. Using the parameters for the rigid H 2 molecule examined below (b = 0.766 64 Å, m = 1.673 72 × 10 −24 kg), k h ranges from 0.0458 at T = 15 K to 1.83 at 600 K. ...
... This effective interaction (as manifest via ψ ik ) obviously is not a simple harmonic, so it is difficult to proceed in an exact analytic manner as we did for the adjacent-image case. However, we can perform a second-order series expansion of ln F (2) eff in terms of the ik distance d ik = 2b sin(ψ ik /2), to identify an effective harmonic interaction, ...
Article
We develop an orientation sampling algorithm for rigid diatomic molecules, which allows direct generation of rings of images used for path-integral calculation of nuclear quantum effects. The algorithm treats the diatomic molecule as two independent atoms as opposed to one (quantum) rigid rotor. Configurations are generated according to a solvable approximate distribution that is corrected via the acceptance decision of the Monte Carlo trial. Unlike alternative methods that treat the systems as a quantum rotor, this atom-based approach is better suited for generalization to multi-atomic (more than two atoms) and flexible molecules. We have applied this algorithm in combination with some of the latest ab initio potentials of rigid H2 to compute fully quantum second virial coefficients, for which we observe excellent agreement with both experimental and simulation data from the literature.
... Approximants have been used to examine a single path of approach for lattice models (viz., along the critical isochore) 7,14-16 and for molecular model fluids (along the critical isotherm). 17 Here, we are interested in an analytic continuation of the virial series that incorporates the first three scaling paths given in Eq. (2), so that we may obtain an equation of state valid at low density, the critical region, and intermediate regions. Care is required to ensure that singularities enforced at the critical point do not introduce anomalies away from the critical region. ...
... Although several choices lead to a convergent description of the SW fluid, we use f (θ) = θ in what follows, as this leads to a desirable scaling property (discussed below). The choice of the r-dependence in the auxiliary function given in Eq. (4) allows the approximant to reduce to a critical isotherm approximant when T = T c , similar to the one given in Ref. 17, such that P = P c − A(ρ)(1 − ρ/ρ c ) δ . The approximant given by Eq. (4) at T = T c differs from the one in Ref. 17 in that it relies on knowledge of B 0 , and is not restricted to ρ < ρ c . ...
... The choice of the r-dependence in the auxiliary function given in Eq. (4) allows the approximant to reduce to a critical isotherm approximant when T = T c , similar to the one given in Ref. 17, such that P = P c − A(ρ)(1 − ρ/ρ c ) δ . The approximant given by Eq. (4) at T = T c differs from the one in Ref. 17 in that it relies on knowledge of B 0 , and is not restricted to ρ < ρ c . ...
Article
Full-text available
The mathematical structure imposed by the thermodynamic critical point motivates an approximant that synthesizes two theoretically sound equations of state: the parametric and the virial. The former is constructed to describe the critical region, incorporating all scaling laws; the latter is an expansion about zero density, developed from molecular considerations. The approximant is shown to yield an equation of state capable of accurately describing properties over a large portion of the thermodynamic parameter space, far greater than that covered by each treatment alone.
... Here, we implement the recent method of asymptotic approximants to analytically continue the power series solution, and thereby construct a highly accurate closed-form solution to (1.1). Asymptotic approximants are used to sum divergent series and may be constructed when asymptotic behaviors are known in two different regions of a domain; implementation details are given in (12)(13)(14)(15)(16)(17). The method is a generalization of two well-known mathematical techniques: asymptotic matching and Padé approximants (18). ...
... Additionally, analytic solutions preserve accuracy when integration or differentiation is required to obtain auxiliary properties of the flow field. The main goal here, however, is to advance the method of asymptotic approximants by demonstrating that the method disclosed in previous work (12)(13)(14)(15)(16)(17) may be applied to (1.1). The ability of the approximant to capture the full range of solutions (as wedge angle is varied), and the accuracy of auxiliary properties obtained using the approximant, are examined as well. ...
... Asymptotic approximants provide nearly exact closed-form solutions to the Falkner-Skan boundary layer equation for varying wedge angle. This adds to the increasing number of problems in disparate areas of mathematical physics to which asymptotic approximants have been applied successfully (12)(13)(14)(15)(16)(17). Advantages of asymptotic approximants, specifically for the Falkner-Skan problem and in general for other problems, are their simple form, ability to yield highly accurate solutions, accuracy in solution derivatives and low computational load. ...
Article
We demonstrate that the asymptotic approximant applied to the Blasius boundary layer flow over a flat plat (Barlow et al., Q. J. Mech. Appl. Math. 70 (2017) 21–48.) yields accurate analytic closed-form solutions to the Falkner–Skan boundary layer equation for flow over a wedge having angle βπ/2 to the horizontal. A wide range of wedge angles satisfying β∈[−0.198837735,1] are considered, and the previously established non-unique solutions for β<0 having positive and negative shear rates along the wedge are accurately represented. The approximant is used to determine the singularities in the complex plane that prescribe the radius of convergence of the power series solution to the Falkner–Skan equation. An attractive feature of the approximant is that it may be constructed quickly by recursion compared with traditional Padé approximants that require a matrix inversion. The accuracy of the approximant is verified by numerical solutions, and benchmark numerical values are obtained that characterize the asymptotic behavior of the Falkner–Skan solution at large distances from the wedge.
... Baker & Gammel's statement may be extended to asymptotic behaviors beyond integer power laws using approximants other than Padés [16,17,18]. An example of such an approximant is found in the review by Frost and Harper [10], used to find an approximate solution for the drag coefficient on a sphere in fluid-filled tube; the approximant they used incorporates the correct non-integer power-law asymptotic behavior. ...
... Approximants are closed form functions that may be analytically differentiated to obtain auxiliary properties without the loss of accuracy that occurs with discretized solutions. Also, physically relevant properties may be cast as unknowns within an approximant, and the approximant can be used as a predictor for such properties [19,16,17,18]. This feature of approximants provides a significant problem-solving advantage. ...
... Padé and other approximants have long been used to predict critical properties of lattice models [28,29,6,7]. With the advent of asymptotic approximants, this approach has recently yielded accurate predictions of critical properties of model fluids [17,18]. The prediction begins by constructing the approximant in the same manner as the previous section. ...
Article
Full-text available
A compact and accurate solution method is provided for problems whose infinite power series solution diverges and/or whose series coefficients are only known up to a finite order. The method only requires that either the power series solution or some truncation of the power series solution be available and that some asymptotic behavior of the solution is known away from the series' expansion point. Here, we formalize the method of asymptotic approximants that has found recent success in its application to thermodynamic virial series where only a few to (at most) a dozen series coefficients are typically known. We demonstrate how asymptotic approximants may be constructed using simple recurrence relations, obtained through the use of a few known rules of series manipulation. The result is an approximant that bridges two asymptotic regions of the unknown exact solution, while maintaining accuracy in-between. A general algorithm is provided to construct such approximants. To demonstrate the versatility of the 1 method, approximants are constructed for three nonlinear problems relevant to mathematical physics: the Sakiadis boundary layer, the Blasius boundary layer, and the Flierl-Petviashvili monopole. The power series solution to each of these problems is underspecified since, in the absence of numerical simulation, one lower-order coefficient is not known; consequently, higher-order coefficients that depend recur-sively on this coefficient are also unknown. The constructed approx-imants are capable of predicting this unknown coefficient as well as other important properties inherent to each problem. The approxi-mants lead to new benchmark values for the Sakiadis boundary layer and agree with recent numerical values for properties of the Blasius boundary layer and Flierl-Petviashvili monopole.
... The constants  A > 0,   A , and 2 0 r entering in (20) will be determined from the integral conditions to which the correlation functions must satisfy. ...
... between the remaining two 9 quantities. If we substitute this relation and the found value of 0 r into expressions (20) and (24), take into account (25) and use the requirement (23), then as a result we obtain an equation for determining the constant   A > 0, after which we also find the constant    A . Within this scheme, using a simple numerical procedure, the following values of the dimensionless constants were found: ...
... Herewith, we used only the theoretically calculated values of the two critical exponents (the leading [10] and the next to the leading (15)), describing the asymptotic decay of ) (r h c , and the exact integral conditions for PCFs. As for the "critical" pressure p c , then in the adopted rough approximation it turns out to be negative: (20) leads to an overestimated contribution of the long-range correlations in  c p , since at large enough distances the interatomic interaction has the character of attraction (cf. [15], where the question of the negative sign of the singular contribution to the "critical" pressure is discussed). ...
Preprint
Using the approach formulated in the previous papers of the author, a consistent procedure is developed for calculating non-classical asymptotic power terms in the total and the direct correlation functions of a critical fluid. Analyzing the Ornstein-Zernike equation allows us to find, for the first time, the values of transcendental exponents 1.73494 and 2.26989 which determine the asymptotic terms next to the leading one in the total correlation function. It is shown that already the simplest approximation based on only two asymptotic terms leads to the correlation functions, which are quantitatively close to the corresponding ones of the Lennard-Jones fluid (argon) in the near-critical state. The obtained results open a way for consistent theoretical interpretation of the experimental data on the critical characteristics of real substances. Both the theoretical arguments and analysis of published data on the experimentally measured critical exponents of real fluids lead to the conclusion that the known assumption of the sameness of the critical characteristics of the Ising model and the fluid in the vicinity of critical point (the universality hypothesis) should be questioned.
... Here, we implement the recent method of asymptotic approximants to analytically continue the power series solution, and thereby construct a highly accurate closed form solution to (1). Asymptotic approximants, originally developed for divergent series problems of thermodynamics [12,13,14], may be constructed when asymptotic behaviors are known in two different regions of a domain; implementation details are given in [15]. The method is a generalization of two well-known mathematical techniques: asymptotic matching and Padé approximants [16]. ...
... Asymptotic approximants provide nearly exact closed-form solutions to the Falkner-Skan boundary layer equation for varying wedge angle. This adds to the increasing number of problems in disparate areas of mathematical physics to which asymptotic approximants have been applied successfully [12,13,14,15,18,17]. Advantages of asymptotic approximants, specifically for the Falkner-Skan problem and in general for other problems, are their simple form, ability to yield highly accurate solutions, accuracy in solution derivatives, and low computational load. ...
Preprint
Full-text available
We demonstrate that the asymptotic approximant applied to the Blasius boundary layer flow over a flat plat (Barlow et al., 2017 Q. J. Mech. Appl. Math., 70(1): 21-48) yields accurate analytic closed-form solutions to the Falkner-Skan boundary layer equation for flow over a wedge having angle $\beta\pi/2$ to the horizontal. A wide range of wedge angles satisfying $\beta\in[-0.198837735, 1]$ are considered, and the previously established non-unique solutions for $\beta<0$ having positive and negative shear rates along the wedge are accurately represented. The approximant is used to determine the singularities in the complex plane that prescribe the radius of convergence of the power series solution to the Falkner-Skan equation. An attractive feature of the approximant is that it may be constructed quickly by recursion compared with traditional Pad\'e approximants that require a matrix inversion. The accuracy of the approximant is verified by numerical solutions, and benchmark numerical values are obtained that characterize the asymptotic behavior of the Falkner-Skan solution at large distances from the wedge.
... It appears that appropriate summations can even capture the correct behavior on approach to the critical point. 4 Where things have broken down is below the critical temperature in fluids that exhibit a vapor-liquid transition. In this regard, it is worth mentioning recent investigations 5 suggesting that this breakdown can be addressed via an Ndependent virial series that avoids summing an infinite number of terms. ...
... This "plateau shoulder" is an anomaly specific to SW fluids, 26 so we choose not to dwell on it at this time. Nevertheless, our analysis suggests that the plateau shoulder may be estimated more precisely with higher virial coefficients, and the qualitatively sensible behavior of A 2 approaching zero at very high densities is guaranteed by the factor of (1 − η) 4 . This behavior is supported in part by Pavlyukin's analysis and by the expectation that fluctuations should diminish as fluids approach their jamming density. ...
Article
Cluster integrals are evaluated for the coefficients of the combined temperature- and density-expansion of pressure: Z = 1 + B 2(β) η + B 3(β) η 2 + B 4(β) η 3 + ⋯, where Z is the compressibility factor, η is the packing fraction, and the Bi (β) coefficients are expanded as a power series in reciprocal temperature, β, about β = 0. The methodology is demonstrated for square-well spheres with λ = [1.2-2.0], where λ is the well diameter relative to the hard core. For this model, the Bi coefficients can be expressed in closed form as a function of β, and we develop appropriate expressions for i = 2-6; these expressions facilitate derivation of the coefficients of the β series. Expanding the Bi coefficients in β provides a correspondence between the power series in density (typically called the virial series) and the power series in β (typically called thermodynamic perturbation theory, TPT). The coefficients of the β series result in expressions for the Helmholtz energy that can be compared to recent computations of TPT coefficients to fourth order in β. These comparisons show good agreement at first order in β, suggesting that the virial series converges for this term. Discrepancies for higher-order terms suggest that convergence of the density series depends on the order in β. With selection of an appropriate approximant, the treatment of Helmholtz energy that is second order in β appears to be stable and convergent at least to the critical density, but higher-order coefficients are needed to determine how far this behavior extends into the liquid.
... Finally, when expressed in terms of a molecular model, with coefficients computed via cluster integrals, the VEOS in principle captures all features related to fluctuations. Indeed, its fundamentally non-classical character appears to make it amenable to resummation using approximants formulated to express known critical behavior [18,19]. ...
Article
Recently reported virial coefficients for the Lennard-Jones model are extrapolated to very high order, and the results are used to study the behavior of virial equation of state (VEOS). Convergence of the VEOS is examined in the context of gas-phase metastability and condensation. Comparison to molecular simulation data shows that the VEOS can accurately describe the equation of state over much of the metastable region, and the stability limits of very low-order isotherms correspond well with simulation-based estimates of the spinodal densities. However, as higher-order terms are added to the density series, the VEOS becomes less capable of characterizing metastable states, and instead appears to be moving toward a description of condensation. The fully-summed VEOS based on virial coefficients extrapolated to infinite order abruptly ends in the metastable region with a branch-point singularity. This form represents the culmination of a sequence of curves in which the pressure reaches a maximum before turning downward, both more sharply and at lower density with increasing series order; the corresponding sequence of maxima converge to the point where the fully-summed VEOS diverges. Thus, the extrapolation-based fully-summed VEOS exhibits the qualitative features of condensation, but it fails to provide quantitative agreement with condensation densities established by molecular simulation. The shortcomings point to a need to better understand the behavior of the virial coefficients with increasing order, perhaps with consideration of the volume dependence of the cluster integrals on which the VEOS is based.
... Alternatively, we can continue to work toward ways to improve convergence for a given number of coefficients. We have developed approximants for pure fluids toward this end, 27,32 and additional development is needed to extend these approaches to mixtures. ...
Article
We report virial coefficients up to sixth order in density for N2, O2, NH3, and CO2, covering temperatures from 50 to 1,000 K. The reported values include coefficients and their first three temperature derivatives, for the pure species as well as all of those needed to evaluate full composition dependence of mixtures formed from any or all of these compounds. The values are obtained by calculation of appropriate cluster integrals using Mayer sampling Monte Carlo, with intermolecular interactions described by the Transferable Potential for Phase Equilibria (TraPPE) force field. All coefficients are fit as a function of temperature, yielding a thermodynamic model with analytic dependence on temperature, density, and composition. The coefficients and properties computed from them are compared to experimental data where available.
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Mixture virial coefficients to seventh order are presented for the system CO2/CH4 at four supercritical temperatures: 323.15, 373.15, 473.15, and 573.15 K. Values are evaluated via the Mayer sampling Monte Carlo method using a three-site TraPPE model for CO2 and a one-site model for CH4. The coefficients are used to compute seven thermodynamic properties (viz., compressibility factor, isothermal compressibility, volume expansivity, isochoric and isobaric heat capacities, Joule-Thomson coefficient, and speed of sound) as a function of mole fraction and density for these temperatures. Comparison is made with corresponding data in the literature as obtained by molecular dynamics simulation, covering densities up to about twice the critical density. Key conclusions are as follows, noting that some exceptions are observed in each case: (a) The virial equation of state (VEOS) to fourth or fifth order describes all properties to within the simulation uncertainty for densities up to at least the critical density, and the addition of terms up to seventh order extends this range considerably. (b) The accuracy of the VEOS is severely diminished for conditions approaching the critical point (the present work extends down to a reduced temperature of 1.06 for CO2), and the study of the pure component behavior suggests the critical singularity blocks convergence for conditions at considerably higher temperatures, albeit at correspondingly higher pressures. (c) Comparison of the VEOS at different orders provides a reliable guide to its accuracy at a given order, so the VEOS can provide a self-assessment of its accuracy when independent data for comparison are unavailable. (d) The VEOS provides a good description of the Joule-Thomson coefficient, including the inversion point in particular. The third-order series is needed to obtain behavior that is qualitatively correct, and the addition of higher-order terms steadily improves the accuracy quantitatively. (e) Under conditions where the seventh-order series is converged, properties can be computed to a given precision with VEOS using much less computational effort in comparison to molecular simulation. (Figure Presented).
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Using the approach formulated in the previous papers of the author, a consistent procedure is developed for calculating non-classical asymptotic power terms in the total correlation function hc(r) of a “critical” fluid. Analyzing the Ornstein-Zernike equation with taking account of the contribution ˜hc4(r) in the direct correlation function cc(r) allows us to find, for the first time, the values of transcendental exponents n′=1.73494... and n″=2.26989...which determine the asymptotic terms next to the leading one in hc(r). It is shown that already the simplest approximation based on only two asymptotic terms, ˜r−6/5(it was found earlier) and ˜r−n′, leads to the functions hc(r) and cc(r), which agree (at least, qualitatively) with the corresponding correlation functions of the Lennard-Jones fluid (argon) in the near-critical state. The obtained results open a way for consistent theoretical interpretation of the experimental data on the critical characteristics of real substances. Both the theoretical arguments and analysis of published data on the experimentally measured critical exponents of real fluids lead to the conclusion that the known assumption of the similarity of the critical characteristics of the Ising model and the fluid in the vicinity of critical point (the “universality hypothesis”) should be questioned.
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We report on our work towards establishing the necessary and suﬃcient conditions for an asymptotically consistent approximant used to predict critical ﬂuid properties based solely on the virial equation of state (virial series) and knowledge of the established universal critical behavior at the critical point. Given enough coeﬃcients, accurate estimates of the critical temperature Tc and pressure Pc of a ﬂuid can be extracted from the virial series. However, using this approach, the critical density ρc is consistently predicted as being too low. We provide evidence that this is a result of the non-singular nature of the virial series. In an eﬀort to explore techniques that would allow one to capture a more accurate estimate of ρc, we draw from known critical scaling-laws and cast the critical density as a branch-point singularity, using an approximant method to analytically continue the series. As a result, critical isotherms are constructed that capture the non-classical (singular) behavior at the critical point, while still retaining the low density behavior of the virial series. The approach to the critical region (from above, T >Tc) is also explored, using approximants that are asymptotically consistent with universal scaling laws.
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Critical phenomena in fluid and fluid mixtures have been the subject of many theoretical and experimental studies during the past decades as has been elucidated in various reviews.1–15 The most striking result of these studies has been the discovery of critical-point universality. Universality of cr...
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We report numerical results for the third virial coefficient of two center Lennard-Jones quadrupolar molecules. Calculations are performed for 35 models with different elongations and quadrupoles over a temperature range from half to twice the critical temperature. It is found that increasing the elongation at fixed quadrupole has the effect of increasing B3. On the other hand, at fixed elongation B3 first decreases with increasing quadrupole at low temperatures, then increases with increasing quadrupole at higher temperatures. We estimate the temperature at which the third virial coefficient vanishes. Although both this temperature and the critical temperature increase with the quadrupole moment, their ratio remains almost constant. We predict the critical properties using two different truncated virial series. The first one employs the exact second and third virial coefficients. The second one approximates the fourth order contribution by using estimates obtained for hard diatomics. It is found that both methods yield fairly good predictions, with a somewhat better performance of the approximate fourth order expansion. The two methods are complementary, however, because they consistently bracket the exact value as determined from computer simulations.
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Virial coefficients for various molecular models are calculated up to B 8 using the Mayer Sampling Monte Carlo method and implemented on a graphics processing unit (GPU). The execution time and performance of these calculations is compared with equivalent computations done on a CPU. The speedup between virial coefficient computations on a CPU (w/optimized C code) and a GPU (w/CUDA) is roughly two orders of magnitude. We report values of B 6, B 7, and B 8 of the Lennard-Jones (LJ) model, as computed on the GPU, for temperatures T = 0.6 to 40 (in LJ units).
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We present results for the fourth virial coefficient of quadrupolar Lennard-Jones diatomics for several quadrupole moments and elongations. The coefficients are employed to predict the critical properties from two different truncated virial series. The first one employs the exact second and third virial coefficients, calculated in our previous work. The second includes also the exact fourth virial coefficient as obtained in this work. It is found that the first method yields already fairly good predictions. The second method significantly improves on the first one, however, yielding good results for both the critical temperature and pressure. Particularly, when compared with predictions from perturbation theories available in the literature, the virial series to fourth order compares favorably for the critical temperature. The results suggest that the failure of perturbation theories to predict the critical temperature and pressure is not only related to the neglect of density fluctuations, but also to poor prediction of the virial coefficients. © 2003 American Institute of Physics.
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Monte Carlo simulations in the grand canonical ensemble were used to obtain liquid-vapor coexistence curves and critical points of the pure fluid and a binary mixture of Lennard-Jones particles. Critical parameters were obtained from mixed-field finite-size scaling analysis and subcritical coexistence data from histogram reweighting methods. The critical parameters of the untruncated Lennard-Jones potential were obtained as Tc∗ = 1.3120±0.0007, ρc∗ = 0.316±0.001 and pc∗ = 0.1279±0.0006. Our results for the critical temperature and pressure are not in agreement with the recent study of Caillol [J. Chem. Phys. 109, 4885 (1998)] on a four-dimensional hypersphere. Mixture parameters were ϵ1 = 2ϵ2 and σ1 = σ2, with Lorentz–Berthelot combining rules for the unlike-pair interactions. We determined the critical point at T∗ = 1.0 and pressure-composition diagrams at three temperatures. Our results have much smaller statistical uncertainties relative to comparable Gibbs ensemble simulations. © 1998 American Institute of Physics.
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This unique book gives a comprehensive account of new mathematical tools used to solve polygon problems. In the 20th and 21st centuries, many problems in mathematics, theoretical physics and theoretical chemistry and more recently in molecular biology and bio-informatics can be expressed as counting problems, in which specified graphs, or shapes, are counted. One very special class of shapes is that of polygons. These are closed, connected paths in space. We usually sketch them in two-dimensions, but they can exist in any dimension. The typical questions asked include "how many are there of a given perimeter?", "how big is the average polygon of given perimeter?", and corresponding questions about the area or volume enclosed. That is to say "how many enclosing a given area?" and "how large is an average polygon of given area?" Simple though these questions are to pose, they are extraordinarily difficult to answer. They are important questions because of the application of polygon, and the related problems of polyomino and polycube counting, to phenomena occurring in the natural world, and also because the study of these problems has been responsible for the development of powerful new techniques in mathematics and mathematical physics, as well as in computer science. These new techniques then find application more broadly. The book brings together chapters from many of the major contributors in the field. An introductory chapter giving the history of the problem is followed by fourteen further chapters describing particular aspects of the problem, and applications to biology, to surface phenomena and to computer enumeration methods.
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This chapter reviews the current status of equations of state for fluids and fluid mixtures in the critical region. Critical phenomena in fluids have been the subject of many theoretical and experimental studies during the past thirty years. The most striking result of these studies has been the discovery of critical-point universality: the microscopic structure of fluids becomes unimportant near a critical point. Moreover, the principle of critical-point universality has been extended to near-critical binary mixtures (isomorphism approach). These discoveries make it possible to develop universal equations of state for fluids in the critical region and enable us to look into the problem of formulating global equations of state for dense fluids and fluid mixtures from an entirely new point of view. The principle of critical-point universality finds its physical origin in the phenomenon that long-range fluctuations of the order parameter (density in one-component fluids or/and concentration in fluid mixtures) dominate in the critical region and that the range of these fluctuations becomes much larger than any other microscopic scale. The spatial extent of these critical fluctuations is determined by a correlation length ξ, which diverges at the critical point. Therefore, the behavior of the thermodynamic properties becomes singular at the critical point. The mathematical nature of the asymptotic, singular, critical behavior is now well understood: it can be characterized by scaling laws with universal critical exponents.
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The paper presents a method for interpolation of the temperature dependence of the coefficients appearing in the virial equation of state. Regression of available data for a virial coefficient at high and low temperatures is used to formulate an approximant that captures the order-of-magnitude behaviour in these extremes. This behaviour is factored out of the data, and the resulting residual is subject to spline interpolation. The value of the virial coefficient at an interpolated temperature is recovered by multiplying by the approximant evaluated at the temperature of interest. The scheme is demonstrated through application to virial data for several model systems, and is shown to provide much more reliable results than direct interpolation performed on the unscaled virial-coefficient data.
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For geothermal applications, a scaled fundamental equation has been formulated in order to represent and tabulate the thermodynmamic properties of isobutane in the critical region. In the supercritical range, the surface joins smoothly with that of M. Waxman and J. S. Gallagher, to which it is a complement. The range of the surface is 405-438 K in temperature, 150-290 kg/m**3 in density. The critical constants are T//c equals 407. 84 plus or minus 0. 02 K, rho //c equals 225. 5 plus or minus 2 kg/m**3, P//c equals 3. 629 plus or minus 0. 002 MPa. Comparisons are made with the PVT data of Beattie et al. , and of M. Waxman, and also with the formulations of M. Waxman and J. S. Gallagher, and of R. D. Goodwin and W. M. Haynes.
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A generalization of the parametric representation for thermodynamic scaling is proposed, introducing a new critical exponent ε. Expansions about the critical point are deduced for the fluids, and to lowest order the asymptotic power-low forms are recovered. An exponent 1-α′ is obtained for the diameter of the coexistence curve. Experimental data are shown to support the predicted forms.
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An analysis is presented of the thermodynamic properties of D2O in the critical region. It is shown that the data can be represented by the same revised and extended scaled fundamental equation formulated earlier for the thermodynamic properties of H2O in critical region. The equation is valid in the range 220–465 kg/m3 in density and 638–683 K in temperature. Tabulated values of the thermodynamic properties of D2O in the critical region are presented. A comparison with a comprehensive analytic fundamental equation, recently formulated by Hill and co-workers, is included in the paper.
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The paper is concerned with some theoretical and empirical attempts to obtain a unified equation for the thermodynamic properties of fluids that incorporates the nonclassical scaling laws near the critical point and crosses over to a classical analytic equation far away from the critical point. Specifically we investigate to what extent the various proposed crossover models agree with the known theoretical predictions for the universal ratios among the asymptotic and correction-to-scaling amplitudes in the power-law expansions of the thermodynamic properties around the critical point.
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We review the existing simulation data and equations of state for the Lennard-Jones (LJ) fluid, and present new simulation results for both the cut and shifted and the full LJ potential. New parameters for the modified Benedict-Webb-Rubin (MBWR) equation of state used by Nicolas, Gubbins, Streett and Tildesley are presented. In contrast to previous equations, the new equation is accurate for calculations of vapour-liquid equilibria. The equation also accurately correlates pressures and internal energies from the triple point to about 4·5 times the critical temperature over the entire fluid range. An equation of state for the cut and shifted LJ fluid is presented and compared with the simulation data of this work, and previously published Gibbs ensemble data. The MBWR equation of state can be extended to mixtures via the van der Waals one-fluid theory mixing rules. Calculations for binary fluid mixtures are found to be accurate when compared with Gibbs ensemble simulations.
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On the basis of the Padé approximant method we deduce from the exact series expansions for the Ising model that the reduced magnetic susceptibility behaves at the critical point as χfcc≈[0.09923/(0.101767-w)]5/4, χbcc≈[0.152773/(0.1561789-w)]5/4, χsc≈[0.22138/(0.218156-w]5/4, χt≈[0.2432/(2-√3-w)]7/4, χsq≈[0.35724/(√2-1-w)]7/4, and χh≈[0.4506/(1/√3-w)]7/4, where w=tanh(J/kT) and the last figure quoted is somewhat uncertain. The spontaneous magnetization is found to behave as (I0/I∞)fcc≈[12.5(0.664658-z2)]0.3, (I0/I∞)bcc≈[10.4(0.5326607-z2)]0.3, (I0/I∞)sc≈[10.9(0.411940-z2)]0.3, where z=exp(-2J/kT) and again the last place quoted is somewhat uncertain. The numbers 5/4 and 7/4 have an error of at most 10-3, and 0.3 of at most 10-2. The lattices referred to are fcc, face-centered cubic; bcc, body-centered cubic; sc, simple cubic; t, triangular; sq, simple quadratic; and h, honeycomb.
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Using recently calculated 8th virial coefficients for hard spheres and disks, singularities in the virial expansion are reinvestigated. Using a Padé analysis, it is concluded that the singularities are associated with crystalline close‐packing and not random close‐packing nor with the fluid→solid phase transition. Accurate equations of state are developed for hard sphere and disk fluids. Estimates of 9th and higher virial coefficients are also given.
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We present Mayer-sampling Monte Carlo calculations of the quantum Boltzmann contribution to the virial coefficients B(n), as defined by path integrals, for n = 2 to 4 and for temperatures from 2.6 K to 1000 K, using state-of-the-art ab initio potentials for interactions within pairs and triplets of helium-4 atoms. Effects of exchange are not included. The vapor-liquid critical temperature of the resulting fourth-order virial equation of state is 5.033(16) K, a value only 3% less than the critical temperature of helium-4: 5.19 K. We describe an approach for parsing the Boltzmann contribution into components that reduce the number of Mayer-sampling Monte Carlo steps required for components with large per-step time requirements. We estimate that in this manner the calculation of the Boltzmann contribution to B(3) at 2.6 K is completed at least 100 times faster than the previously reported approach.
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Wertheim's multi-density formalism for pair-wise additive molecular interaction is extended to handle non-additive contributions and is applied to formulate an equation of state (WEOS) for the Gaussian-Charge Polarizable Model (GCPM) of water, with cluster integrals appearing in the theory calculated via the Mayer sampling Monte Carlo method. At both sub- and super-critical temperatures, the equation of state of GCPM water obtained from WEOS converges well to Monte Carlo simulation data, and performs significantly better than the conventional virial treatment (VEOS). The critical temperature for GCPM water using a 4th-order WEOS is given to within 1.3% of the established value, compared to a 17% error shown by 5th-order VEOS; as seen in previous applications, the critical density obtained from both VEOS and WEOS significantly underestimates the true critical density for GCPM water. Examination of the magnitudes of the computed cluster diagrams at the critical density finds that negligible contributions are made by clusters in which a water molecule has both of its hydrogens involved in association interactions.
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We report values of the virial coefficients B n of the Lennard-Jones (LJ) model, as computed by the Mayer Sampling Monte Carlo method. For n = 4 and 5, values are reported for 103 temperatures T = 0.62 to 40.0 (in LJ units); for n = 6, 31 values are reported for T = 0.625 to 20.0; for n = 7, 15 values are reported from T = 0.625 to 10; and for n = 8, four values are reported from T = 0.75 to 10. Data are used to estimate the location of the LJ critical point, and the critical temperature estimated this way is given to within 0.8% of the established value, while the critical density is too low by 10%. Data derived from the virial equation of state (VEOS) are compared to pressures and internal energies calculated by Monte Carlo simulation. Simulations of systems ranging from 125 to 30,000 particles are extrapolated to infinite system size, and it is shown that the VEOS–when applied at densities where the series has reached convergence–provides results closer to the infinite-system values than obtained by any of the finite-system simulations. For n = 6, convergence of VEOS (within a 1% tolerance) is obtained for densities up to the spinodal for subcritical temperatures and up to ρ = 0.4 (in LJ units) in the vicinity of the critical temperature; the range of applicability of VEOS increases with temperature, reaching for example densities of 0.65 for T = 5.0 and 0.8 for T = 8.0 when truncated at n = 6.
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We present a method in which the Padé approximant is used to calculate the asymptotic behavior of a function from the first few terms of its power series. We illustrate this method by two examples and give a partial justification of it. One of the examples is a calculation of the binding energy of a Fermi gas of hard spheres.
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Several methods of extrapolating the virial coefficients, including those proposed in this work, are discussed. The methods are demonstrated on predicting higher virial coefficients of one-component hard spheres. Estimated values of the eleventh to fifteenth virial coefficients are suggested. It has been speculated that the virial coefficients, B_n, beyond B_{14} may decrease with increasing n, and may reach negative values at large n. The extrapolation techniques may be utilized in other fields of science where the art of extrapolation plays a role.
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We evaluate the virial coefficients Bk for k £ 10k\leq 10 for hard spheres in dimensions D=2,¼,8.D=2,\cdots,8. Virial coefficients with k even are found to be negative when D ³ 5.D\geq 5. This provides strong evidence that the leading singularity for the virial series lies away from the positive real axis when D ³ 5D\geq 5 . Further analysis provides evidence that negative virial coefficients will be seen for some k > 10 for D = 4, and there is a distinct possibility that negative virial coefficients will also eventually occur for D = 3.
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The analysis of fluid criticality in the framework of the “virial” approach leads, usually, to pure classical results. In the present paper we propose a consistent procedure allowing us to find the non-classical critical parameters of real fluid starting from the representation of the pressure near the critical point as the sum of a regular (a few of the first terms of the virial series) and a singular (non-analytical “remainder” of the series) part. The critical temperature, density, and the singular contribution to the fluid pressure are found self-consistently using not two(by van der Waals) but three(as in the fluctuation theory) conditions on the density partial derivatives of the pressure at the critical point. The calculated critical parameters converge rather rapidly to some limiting values as the number of terms in the regular part of the representation of the pressure is increased. Our calculations (when taking account of the virial terms up to the sixth one, inclusively) are in accordance, on the whole, with the numerical “experiments” data for the Lennard-Jones fluid, although we predict a somewhat greater (approximately 10%) value of the critical density. The refinement of the obtained results can be achieved when using more precise values of the higher (i.e. the seventh onwards) virial coefficients.
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A brief account is given of a number of generalisations of Pad approximants defined and studied over the past twenty years. These generalisations include multi-point approximants, approximants based upon differential equations, multivalued approximants, multivariate approximants, and approximants defined from series of orthogonal functions. The general class of Hermite-Pad approximants is discussed, and various applications are noted.
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A new analytic equation of state for the Lennard-Jones fluid is proposed. The equation is based on a perturbed virial expansion with a theoretically defined temperature-dependent reference hard sphere term. The expansion is written for the Helmholtz free energy which guarantees the thermodynamic consistency of the pressure and internal energy. The equation covers much wider range of temperatures (up to seven times the critical temperature) than existing equations and is significantly more accurate and has less parameters than the best equation available to date, the modified Benedict-Webb-Rubin equation due to Johnson, Zollweg, and Gubbins (1993, Mol. Phys. 78: 591-618). As a side-product, highly accurate explicit analytic correlations of the hard sphere diameters, as given by both the hybrid Barker-Henderson and Weeks-Chandler-Andersen theories, have been obtained.Computer simulation data to be regressed by the equation have been compiled from several sources and critically assessed. It has been shown that many literature data for state points with a large compressibility are subject to large systematic finite-size errors. Additional simulations on a series of systems of different sizes have been therefore performed to facilitate the extrapolation to the thermodynamic limit in the region close to the critical point.
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Long-range critical fluctuations appear to affect the thermodynamic properties of fluids in a very large range of temperatures and densities around the critical point. To treat these effects the universal scaling laws, which are valid in the immediate vicinity of the critical point, need to be extended to account for a crossover to classical thermodynamic behavior far away from the critical point. The paper considers approaches for dealing with this problem and attempts to elucidate the physical features of the crossover from singular asymptotic critical thermodynamic behavior to classical thermodynamic behavior.
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It is well known that classical equations of state, like the Van der Waals equation, fail in the region near the critical point of a fluid, where the behavior of the thermodynamic properties is strongly affected by density fluctuations. In this paper, we present a theoretical approach to correct classical cubic equations of state for the effects of critical fluctuations. The approach utilizes a transformation deduced from the renormalization-group theory of critical phenomena that was developed earlier for a classical Landau expansion of the Helmholtz-free-energy density. Using the Van der Waals equation as an example, we explain how critical fluctuations lower the critical temperature, flatten the coexistence curve, induce a singularity in the isochoric heat capacity, and lead to apparent critical exponents that depend on the temperature distance from the critical point.
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A new set of united-atom Lennard-Jones interaction parameters for n-alkanes is proposed from fitting to critical temperatures and saturated liquid densities. Configurational-bias Monte Carlo simulations in the Gibbs ensemble were carried out to determine the vapor-liquid coexistence curves for methane to dodecane using three united-atom force fields: OPLS [Jorgensen, et al. J. Am. Chem. Sec. 1984, 106, 813], SKS [Siepmann, et al. Nature 1993, 365, 330], and TraPPE. Standard specific densities and the high-pressure equation-of-state for the transferable potentials for phase equilibria (TraPPE) model were studied by simulations in the isobaric-isothermal and canonical ensembles, respectively. It is found that one set of methyl and methylene parameters is sufficient to accurately describe the fluid phases of all n-alkanes with two or more carbon atoms. Whereas other n-alkane force fields employ methyl groups that an either equal or larger in size than the methylene groups, it is demonstrated here that using a smaller methyl group yields a better fit to the set of experimental data. As should be expected from an effective pair potential, the new parameters do not reproduce experimental second virial coefficients. Saturated vapor pressures and densities show small, but systematic deviation from the experimental data.
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The effects of higher-order contributions to the linearized renormalization group equations in critical phenomena are discussed. This analysis leads to three quite different results: (i) An exact scaling law for redefined fields is obtained. These redefined fields are normally analytic functions of the physical fields. Corrections to the standard power laws are derived from this scaling law. (ii) The theory explains why logarithmic terms can exist in the free energy. (iii) The case in which the energy scales like the dimensionality is analyzed to show that quite anomalous results may be obtained in this special situation.
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We report the results from Mayer-sampling Monte Carlo calculations of the virial coefficients of the united-atom TraPPE-UA model of normal alkanes. For alkane chain lengths from n=2 to 20 (where n is the number of carbon atoms), results are given for the virial coefficients B(2), B(3), and B(4); results for B(5) are given for chains up to length n=12; and results for B(6) are given for chains of length n=2, 3, and 4. In all cases, values are given for temperatures ranging from 200 K to 2000 K in 20-50 K increments. The values are used to calculate the equation of state for butane and the pressure-density behavior is compared to experimental data at 350 and 550 K. Critical points are calculated for all systems and compared to simulation data previously taken for the same molecular model, and to experiment. The comparison with temperature is very good (within 1.5% for all chain lengths up to n=12), while the critical density is underestimated by about 5%-15% and the critical pressure is given within about 10%. The convergence behavior of the virial equation of state as applied across the n-alkane series is well characterized by corresponding states, meaning that the accuracy at a given density relative to the critical density does not deteriorate with increasing chain length.
Article
The helium pair potential was computed including relativistic and quantum electrodynamics contributions as well as improved accuracy adiabatic ones. Accurate asymptotic expansions were used for large distances R. Error estimates show that the present potential is more accurate than any published to date. The computed dissociation energy and the average R for the (4)He(2) bound state are 1.62+/-0.03 mK and 47.1+/-0.5 A. These values can be compared with the measured ones: 1.1(-0.2)(+0.3) mK and 52+/-4 A [R. E. Grisenti, Phys. Rev. Lett. 85, 2284 (2000)].