Statistical Mechanical Perturbation Theory Research Articles

Perturbation theory in a microcanonical ensemble

The microcanonical ensemble is a natural starting point of statistical mechanics. However, when it comes to perturbation theory in statistical mechanics, mainly the canonical and grand canonical ensembles have been used. In this article we show how the microcanonical ensemble can be directly used to carry out perturbation theory for both non-interacting and interacting systems. We obtain the first non-trivial order answers for the specific heat of anharmonic oscillators and for the virial expansion in real gases. They are in exact agreement with the results obtained from the canonical ensemble. In addition, we show how crossover functions for the specific heat of anharmonic oscillators can be constructed using a microcanonical ensemble and also how the subsequent terms of the virial expansion can be obtained. We find that if we consider quantum free particles in a one-dimensional box of extension L, then both the ensembles give an unusual result for the first correction to the specific heat in the high temperature limit.

A simple theory for interfacial properties of dilute solutions.

Recent studies suggest that cosolute mixtures may exert significant non-additive effects upon protein stability. The corresponding liquid-vapor interfaces may provide useful insight into these non-additive effects. Accordingly, in this work, we relate the interfacial properties of dilute multicomponent solutions to the interactions between solutes. We first derive a simple model for the surface excess of solutes in terms of thermodynamic observables. We then develop a lattice-based statistical mechanical perturbation theory to derive these observables from microscopic interactions. Rather than adopting a random mixing approximation, this dilute solution theory (DST) exactly treats solute-solute interactions to lowest order in perturbation theory. Although it cannot treat concentrated solutions, Monte Carlo (MC) simulations demonstrate that DST describes the interactions in dilute solutions with much greater accuracy than regular solution theory. Importantly, DST emphasizes a fundamental distinction between the "intrinsic" and "effective" preferences of solutes for interfaces. DST predicts that three classes of solutes can be distinguished by their intrinsic preference for interfaces. While the surface preference of strong depletants is relatively insensitive to interactions, the surface preference of strong surfactants can be modulated by interactions at the interface. Moreover, DST predicts that the surface preference of weak depletants and weak surfactants can be qualitatively inverted by interactions in the bulk. We also demonstrate that DST can be extended to treat surface polarization effects and to model experimental data. MC simulations validate the accuracy of DST predictions for lattice systems that correspond to molar concentrations.

Exploring the Interplay between Disordered and Ordered Oligomer Channels on the Aggregation Energy Landscapes of α-Synuclein.

The abnormal aggregation of α-synulcein is associated with multiple neurodegenerative diseases such as Parkinson's disease. The hydrophobic non-amyloid component (NAC) region of α-synuclein comprises the core of the fibril in vitro and in vivo. In this work, we study the aggregation landscape of the hydrophobic NAC region of α-synuclein using a transferrable coarse-grained force field, the associative memory water-mediated structure, and energy model (AWSEM). Using structural similarity, we can group metastable states on the free energy landscape of aggregation into three types of oligomers: disordered oligomers, prefibrillar oligomers with disordered tips, and ordered prefibrillar oligomers. The prefibrillar oligomers with disordered tips have more in-register parallel β strands than do the fully disordered oligomers but have fewer in-register parallel β strands than the ordered prefibrillar oligomers. Along with the ordered prefibrillar species, the disordered oligomeric states dominate at small oligomer sizes while the prefibrillar species with disordered tips thermodynamically dominate with the growth of oligomers. The topology of the aggregation landscape and observations in simulations suggest there is backtracking between ordered prefibrillar oligomers and other kinds of oligomers as the aggregation proceeds. The significant structural differences between the ordered prefibrillar oligomers and the disordered oligomers support the idea that the growth of these two kinds of oligomers involves kinetically independent parallel pathways. In contrast, the overall structural similarity between the fully ordered prefibrillar oligomers and the prefibrillar oligomers with disordered tips implies that two channels can interconvert on slower time scales. We also evaluate the effects of phosphorylation on the aggregation free energy landscape using statistical mechanical perturbation theory.

Exploring the interplay between fibrillization and amorphous aggregation channels on the energy landscapes of tau repeat isoforms

Filaments made up of different isoforms of tau protein are associated with a variety of neurodegenerative diseases. Filaments made up of the 4R-tau isoform, which has four repeat regions (R1 to R4), are found in patients suffering from Alzheimer's disease, while filaments made of the 3R-tau isoform, which contains only three repeat units (R1, R3, and R4), are found in patients with Pick's disease (frontotemporal dementia). In this work, a predictive coarse-grained protein force field, the associative memory water-mediated structure and energy model (AWSEM), is used to study the energy landscapes of nucleation of the two different fibrils derived from patients with Pick's and Alzheimer's diseases. The landscapes for nucleating both fibril types contain amorphous oligomers leading to branched structures as well as prefibrillar oligomers. These two classes of oligomers differ in their structural details: The prefibrillar oligomers have more parallel in-register β-strands, which ultimately lead to amyloid fibrils, while the amorphous oligomers are characterized by a near random β-strand stacking, leading to a distinct amorphous phase. The landscape topography suggests that there must be significant structural reordering, or "backtracking," to transit from the amorphous aggregation channel to the fibrillization channel. Statistical mechanical perturbation theory allows us to evaluate the effects of changing concentration on the aggregation free-energy landscapes and to predict the effects of phosphorylation, which is known to facilitate the aggregation of tau repeats.

A correlation of densities of liquid polar and nonpolar refrigerants from critical temperature

ABSTRACTWe developed an equation of state based on statistical–mechanical perturbation theory for polar and nonpolar refrigerants. The equation is that the Ihm, Song and Mason, in which the three temperature dependant parameters are, calculated by means of the corresponding states correlations. In this work, it is shown that using the critical constants of fluids as scale constants would correlate the temperature-dependent parameters of the equation of state with sufficient accuracy. We have tested the predicted equation of state against the experimental data for a large number of haloalkanes including a number of the environmentally acceptable HFCs (R23, R32, R41, R125, R134a, R143a, R152a, R227ea, R236ea, R236fa, R245ca, R245fa), the HCFCs (R22, R123, R124, R141b, R142b), the traditional CFCs (R11, R12, R13, R113, R114, R115) and the fluorocarbons refrigerants (R14, R116, R218, RC318). A comparison is also made by other scale parameters based on experimental data.

Song and Mason Equation of State for Refrigerants

In this work, the Song and Mason equation of state has been applied to calculate the PVT properties of refrigerants. The equation of state is based on the statistical-mechanical perturbation theory of hard convex bodies. The theory has considerable predictive power, since it permits the construction of the PVT surface from the normal boiling temperature and the liquid density at the normal boiling point. The average absolute deviation for the calculated densities of 11 refrigerants is 1.1%.

Exploring the aggregation free energy landscape of the amyloid-β protein (1–40)

A predictive coarse-grained protein force field [associative memory, water-mediated, structure, and energy model for molecular dynamics (AWSEM)-MD] is used to study the energy landscapes and relative stabilities of amyloid-β protein (1-40) in the monomer and all of its oligomeric forms up to an octamer. We find that an isolated monomer is mainly disordered with a short α-helix formed at the central hydrophobic core region (L17-D23). A less stable hairpin structure, however, becomes increasingly more stable in oligomers, where hydrogen bonds can form between neighboring monomers. We explore the structure and stability of both prefibrillar oligomers that consist of mainly antiparallel β-sheets and fibrillar oligomers with only parallel β-sheets. Prefibrillar oligomers are polymorphic but typically take on a cylindrin-like shape composed of mostly antiparallel β-strands. At the concentration of the simulation, the aggregation free energy landscape is nearly downhill. We use umbrella sampling along a structural progress coordinate for interconversion between prefibrillar and fibrillar forms to identify a conversion pathway between these forms. The fibrillar oligomer only becomes favored over its prefibrillar counterpart in the pentamer where an interconversion bottleneck appears. The structural characterization of the pathway along with statistical mechanical perturbation theory allow us to evaluate the effects of concentration on the free energy landscape of aggregation as well as the effects of the Dutch and Arctic mutations associated with early onset of Alzheimer's disease.

Prediction of Thermodynamic Properties and Phase Behavior of Fluids and Mixtures with the SAFT-γ Mie Group-Contribution Equation of State

Group contribution (GC) approaches are based on the premise that the properties of a molecule or a mixture can be determined from the appropriate contributions of the functional chemical groups present in the system of interest. Although this is clearly an approximation, GC methods can provide accurate estimates of the properties of many systems and are often used as predictive tools when experimental data are scarce or not available. Our focus is on the SAFT-γ Mie approach [Papaioannou, V.; Lafitte, T.; Avendaño, C.; Adjiman, C. S.; Jackson, G.; Müller, E. A.; Galindo, A. Group contribution methodology based on the statistical associating fluid theory for heteronuclear molecules formed from Mie segments. J. Chem. Phys. 2014, 140, 054107–29] which incorporates a detailed heteronuclear molecular model specifically designed for use as a GC thermodynamic platform. It is based on a formulation of the recent statistical associating fluid theory for Mie potentials of variable range, where a formal statistical–mechanical perturbation theory is used to maintain a firm link between the molecular model and the macroscopic thermodynamic properties. Here we summarize the current status of the SAFT-γ Mie approach, presenting a compilation of the parameters for all functional groups developed to date and a number of new groups. Examples of the capability of the GC method in describing experimental data accurately are provided, both as a correlative and as a predictive tool for the phase behavior and the thermodynamic properties of a broad range of complex fluids.

Extension of Tao-Mason Equation of State to Heavy n-Alkanes

In our previous paper we extended the Tao and Mason equation of state (TM EOS) to refrigerant fluids, using the speed of sound data. This is a continuation for evaluating TM EOS in predicting PVT properties of heavy n-alkanes. Liquid density of long-chain n-alkane systems from C9 to C20 have been calculated using an analytical equation of state based on the statistical-mechanical perturbation theory. The second virial coefficients of these n-alkanes are scarce and there is no accurate potential energy function for their theoretical calculation. In this work the second virial coefficients are calculated using a corresponding state correlation based on surface tension and liquid density at the freezing point. The deviation of calculated densities of these alkanes is within 0.5% from experimental data. The densities of n-alkanes obtained from the TM EOS are compared with those calculated from Ihm-Song-Mason equation of state and the corresponding-states liquid densities (COSTALD). Our results are in favor of the preference of the TM EOS over other two equations of state.

Equation of state for mercury: revisited

An analytical equation of state by Song and Mason is developed to calculate the PVT properties of mercury. The equation of state is based on the statistical-mechanical perturbation theory of hard convex bodies and can be written as a fifth-order polynomial in the density. There exists three temperature-dependent parameters in the equation of state; the second virial coefficient, an effective molecular volume, and a scaling factor for the average contact pair distribution function of hard convex bodies. The temperature-dependant parameters have been calculated using corresponding-states correlations based on the surface tension and the liquid density at the normal boiling point. Employing the present equation of state, we have calculated the PVT properties of mercury over a wide range of temperatures and pressures. The average absolute deviation for the calculated density of mercury in the saturation and compressed states is 1.09%.

Correlation of volumetric properties of binary mixtures of some ionic liquids with alcohols using equation of state

In our previous paper, we extended the Tao and Mason equation of state (TM EOS) to pure ionic liquids. Here we apply TM EOS based on statistical–mechanical perturbation theory to binary mixtures of ionic liquids. Three temperature-dependent quantities are needed to use the equation of state: the second virial coefficient, B2, effective van der Waals co-volume, b, and a scaling factor, α. The second virial coefficients are calculated from a correlation that uses the normal boiling temperature and normal boiling density. α and b can also be calculated from the second virial coefficient by scaling. In this procedure, the number of input parameters, for calculation of B2, α, and b reduced from 5 (i.e., critical temperature, critical pressure, acetric factor, Boyle temperature TB, and the Boyle volume υB) to 2 (i.e., Tbp and ρbp). At close inspection of the deviations given in this work, the TM EOS predicts the densities with a mean AAD of 1.69%. The density of selected system obtained from the TM EOS has been compared with those calculated from perturbed-hard-sphere equation of state. Our results are in favor of the preference of the TM EOS over another equation of state. The overall average absolute deviation for 428 data points that calculated by perturbed-hard-sphere equation of state is 2.60%.

Discotic Nematic-Isotropic Phase Transition Properties for a Disc-Like Mesogen

A statistical mechanical perturbation theory has been used to describe discotic nematic-isotropic phase transition for a discotic liquid crystal, hexa(p-octyl phenyl ethynyl)benzene. We have employed the decoupling approximation, which introduces anisotropy in the pair correlation function, to investigate the discotic nematic-isotropic transition properties. The variation of second rank orientational order parameter with temperature has been computed and it shows qualitatively similar behavior to that of experimental studies of Aver'yanov [1]. However, the numerical value of order parameter at the transition found in the present work is larger than that of Aver'yanov's. The various thermodynamic properties have also been calculated at the transition.

Performance assessment of Tao–Mason equation of state: Results for vapor–liquid equilibrium properties

The present work evaluates the performance of a molecular-based equation of state in predicting thermodynamic properties of several fluids in a very wide range of temperatures encompassing 100 K < T < 1100 K and pressures ranging from zero to 3200 bar. The theoretical equation of state (EOS) is that of Tao–Mason (TM) which is based on statistical mechanical perturbation theory. The 21 fluids including: argon (Ar), krypton (Kr), xenon (Xe), nitrogen (N 2), oxygen (O 2), carbon dioxide (CO 2), methane (CH 4), ethane (C 2H 6), propane (C 3H 8), normal butane (n-C 4H 10), isobutene (i-C 4H 10), ethene (C 2H 4), benzene (C 6H 6), toluene (C 7H 8) as well as refrigerants consisting of 1,1,1,2 tetra fluoroethane (R134a), tetrafluoromethane (R14), chlorodifluromethane (R22), 1,1,1-trifluoroethane (R143a), 1,1,1-trifluoro,2,2-dichloroethane (R123), octafluoropropane (R218), and 1,1-difluoroethane (R152a) are selected and compared with literature data. The calculations cover the ranges from the dilute vapor or gas to the highly compressed liquid and supercritical regions. The thermodynamic properties are the vapor and liquid densities, the vapor pressure, the internal energy, the enthalpy, the entropy, the heat capacity at constant pressure and constant volume, and the speed of sound. It was found that the overall agreement with literature in all phases especially the vapor/gas phase is remarkable. Furthermore, the Zeno line regularity can be well represented by the TM EOS. Finally, the TM EOS is further assessed through comparing with the Ihm–Song–Mason (ISM) equation of state. In general, the TM EOS outperforms the ISM equation of state.

Modification of Tao-Mason equation of state: Application to refrigerant fluids

The present work addresses a simplified procedure for the calculation of certain parameters in the Tao-Mason (TM) equation of state needed to estimate thermodynamic properties of 12 ozone–friendly refrigerants fluids. The calculations cover wide ranges extending from the dilute vapor or gas to highly compressed liquid and supercritical region. The theoretical equation of state (EOS) is that of Tao and Mason which is based on statistical mechanical perturbation theory. The novelty of the present work is that we use two scaling constants, the enthalpy of vaporization and the molar density, both evaluated at the normal boiling temperature. This approach contrasts with the original procedure that employs five scaling constants including the critical temperature, the critical pressure, the Boyle temperature, the Boyle pressure and the acentric factor for calculating three parameters second virial coefficient and temperature-dependent parameters. The modified TM EOS predicts thermodynamic properties with equal or better accuracy than those emerging from the original TM EOS.

Equation of state for thermodynamic properties of pure and mixtures liquid alkali metals

We developed an equation of state based on statistical–mechanical perturbation theory for pure and mixtures alkali metals. Thermodynamic properties were calculated by the equation of state, based on the perturbed-chain statistical associating fluid theory (PC-SAFT). The model uses two parameters for a monatomic system, segment size, σ, and segment energy, ɛ. In this work, we calculate the saturation and compressed liquid density, heat capacity at constant pressure and constant volume, isobaric expansion coefficient, for which accurate experimental data exist in the literatures. Results on the density of binary and ternary alkali metal alloys of Cs–K, Na–K, Na–K–Cs, at temperatures from the freezing point up to several hundred degrees above the boiling point are presented. The calculated results are in good agreement with experimental data.

A new equation of state derived by the statistical mechanical perturbation theory

We have derived an analytical equation of state (EOS) based on the soft-core statistical mechanical perturbation theory for fluids, using the Weeks–Chandler–Andersen (WCA) theory recently developed by Ben–Amotz–Stell (BAS) for the choice of the hard-sphere diameter, but with a new algorithm for calculation of the pair and many-body interactions. We have used Carnahan–Starling expression with the Boltzmann factor criterion (BFC) as an effective hard-sphere diameter for the reference system, and also decomposed the perturbed pair potential to symmetric and asymmetric terms. The former term is due to the many-body interactions at high densities as was used in the linear isotherm regularity known as LIR EOS, and the latter term supports the interaction of two isolated particles that is dominating at low densities. The resulting EOS is obtained as Z = Z cs + αρ + Aρ 3 + Bρ 5, in which Z cs is the Carnahan–Starling expression for the compressibility factor of the reference system which contains the effective van der Waals co-volume, and α is due to the asymmetric interaction term, or the attraction contribution of the second virial coefficient. The A and B parameters are the attractive and repulsive contributions of symmetric term, respectively. The temperature-dependencies of all parameters of the EOS are obtained. We select some different fluids, namely Ar, N 2, CH 4, Ne, CO 2, C 2H 6, C 3H 8, NH 3 and H 2O which are spherical, roughly spherical, non-spherical, polar and associated fluids, due to their abundance of P– V– T experimental data. We have found that, except for the critical region, 0.8 < ρ r ≤ 1.5, 1 ≤ T r ≤ 1.5, the new EOS is accurate for all temperatures and densities available in the literatures, in such a way that the average percent deviation of density for Ar, Ne, N 2, CH 4, CO 2, C 2H 6, C 3H 8, and NH 3 is less than 2.81%. Then some thermodynamic properties including vapor–pressure curve, Joule–Thomson inversion curve, P– T isochors curves, and the second virial coefficient have been applied to check the accuracy of the new EOS. Results for some isochors of argon show that the new EOS gives a small negative curvature for all isochors. The new EOS prediction of the inversion Joule–Thomson curve is reasonable, and its prediction of Clausius–Clapeyron diagram for neon and argon is very accurate, but small deviation for methane and nitrogen can be seen.

Toward a statistical mechanical theory for water: Analytical theory for a short-ranged reference system

Starting from a realistic Hamiltonian and making use of recent findings that the properties of associating fluids are determined primarily by short-ranged interactions, this methodology has been implemented using statistical mechanical approaches and thermodynamic perturbation theory for the TIP4P model of water. We focus on the short-range reference system for which an analytic expression for the Helmholtz free energy is derived. It is found that the model (reference system) exhibits, in addition to a faithful representation of the structure of water, the same features that are characteristic for real water, namely, (i) the temperature of the density maximum and its pressure dependence, including the inflection point at high pressures and (ii) the temperature minima of the constant pressure heat capacity and the coefficient of isothermal compressibility.

Thermodynamic perturbation theory in fluid statistical mechanics

A methodology is proposed that pushes the thermodynamic perturbation theory (TPT) from first order to higher order. The second-order correction is superior to a macroscopic compressibility (MC) approximation of Barker and Henderson. The present third-order TPT performs far better than the original first-order TPT and second-order TPT based on the MC approximation for many subfields in fluid statistical mechanics, such as predicting excess Helmholtz free energy, excess chemical potential, bulk pressure, gas-liquid coexistence, and solid-liquid equilibrium of very short-range potential fluids. A nonuniform version of the TPT is proposed; it is also shown that the nonuniform third-order TPT performs far better than the nonuniform first-order TPT in predicting density profile of fluids in critical region. The present report indicates that the TPT still can be a "universal" and accurate theoretical tool that has general applicability in fluid statistical mechanics, especially in soft-matter physics.

Comment on “Molecular Dynamics Simulation of the Liquid/Vapor Interface of Aqueous Ethanol Solutions as a Function of Concentration”

Molecular dynamics computer simulations are utilized to study the structural and thermodynamic properties of the liquid/vapor interface of aqueous ethanol solutions as a function of concentration. In addition, the free energy profile for inserting a single ethanol molecule into a 0.059 mole fraction aqueous ethanol solution is calculated using statistical mechanical perturbation theory. The calculated free energy for solvation of an ethanol molecule in the bulk solution, the surface tension as a function of ethanol concentration, and the average orientation of ethanol molecules at the solution/vapor interface are in agreement with the corresponding experimental data. The calculated equilibrium free-energy profile, however, exhibits a barrier to solvation that is considerably smaller than that predicted by the resistance model for mass accommodation.

Saturated Liquid Densities for 33 Binary Refrigerant Mixtures Based on the ISM Equation of State

In this work, the ISM equation of state based on statistical-mechanical perturbation theory has been extended to liquid refrigerant mixtures by using correlations of Boushehri and Mason. Three temperature-dependent parameters are needed to use the equation of state: the second virial coefficient, B2(T), an effective van der Waals covolume, b(T), and a scaling factor, α (T). The second virial coefficients are calculated from a correlation based on the heat of vaporization, ΔHvap, and the liquid density at the normal boiling point, ρnb. α(T) and b(T) can also be calculated from second virial coefficients by a scaling rule. The theory has considerable predictive power, since it permits the construction of the PVT surface from the heat of vaporization and the liquid density at the normal boiling point. The equation of state was tested on 33 liquid mixtures from 12 refrigerants. The results indicate that the liquid densities can be predicted to at most 2.8% over a wide range of temperatures, 170–369 K.