Generic Van Der Waals Equation Research Articles

Polymer free volume and its connection to the entanglement length and the plateau modulus via polymer mode-coupling theory and equation of state

Using a many-chain system of Gaussian chains interacting with each other through the Lennard-Jones (LJ) potential, we demonstrate that there exists a connection between polymer fractional free volume (f) and the entanglement chain length (Ne). Here, f is determined by the generic van der Waals (GvdW) equation of state using the intermolecular radial distribution function [g(r)] of the Gaussian chain generated by the polymer reference interaction sites model while Ne by the polymer mode-coupling theory (PMCT) for strongly coupled macromolecules. The key concept in PMCT is that intermolecular forces experienced by a bead in a chain surrounded by other chains are strongly coupled to the site specific intramolecular radial distribution function. The calculated Gp and η of polyethylene with chain lengths from 500 to 2000 interacting with one another through the LJ potential agree well with experiment and molecular dynamics simulation.

An evaluation of the self-diffusion coefficient of liquid hydrogen via the generic van der Waals equation of state and modified free volume theory

The modified free volume (MFV) theory was applied to evaluation of the self-diffusion coefficient of liquid hydrogen in which the nuclear quantum effect appears. The free volume of the hydrogen was defined via the generic van der Waals (GvdW) equation of state, which was derived based on the centroid path integral notation. The computational results of the self-diffusion coefficient were compared to those of the experimental data, centroid molecular dynamics (CMD) methods, and previous studies. This comparison revealed that the self-diffusion coefficient of liquid hydrogen computed by means of the MFV theory reproduces the experimental data with the same precision as CMD and previous studies. In conclusion, even though the MFV theory is a completely classical mechanism, the self-diffusion coefficient of a quantum liquid can be calculated on the basis of the MFV theory by treating the centroid as a representative of the quantum molecule.

Investigation of Thermodynamic Consistency Test of Carbon Dioxide (CO2) in Room-Temperature Ionic liquids using Generic van der Waals Equation of State

Thermodynamic consistency test of isothermal vapor-liquid equilibrium (VLE) data of various binary systems containing Carbon dioxide (CO2)/Room temperature ionic liquids (RTILs) have been investigated in wide ranges of pressures in each isotherm precisely. In this paper Generic van der Waals (GvdW) equation of state (EoS) coupled with modified van der Waals Berthelot mixing rule has successfully been applied for correlating P-T-x binary data. The optimum parameters were obtained by minimizing the average relative deviation between modeled and experimental data based on the bubble pressure algorithm. Modeling is highly shown satisfactory in all cases which means that deviations in correlated data are low subsequently can prove that the flexibility and capability of the proposed model for thermodynamic consistency. Results of the consistency test represented ten isothermal experimental data set to be thermodynamically consistent, fourteen were declared to be not fully consistent and just four isothermal experimental data sets were represented to be thermodynamically inconsistent.

The development of van der Waals equation of state for ionic liquids

The main purpose of this paper is to develop van der Waals equation of state for calculating the PρT properties of some ionic liquids. This equation of state is based on the generic van der Waals equation of state in which the parameters of this equation of state are dependent of density and temperature. In order to improve the predictive power of the mentioned equation of state for calculation of density of ionic liquids, modifications on the parameters of this equation of state have been considered. The ionic liquids under consideration in this work include the cation: imidazolium and anions: tetrafluoroborate, thiocyanate, trifluoromethanesulfonate, ethylsulfate, bis (trifluoromethylsulfonyl) imide and hexafluorophosphate. The densities of these compounds were calculated from the modified equation of state at different temperatures and pressures. We will see that the comparison of predicted densities from the modified equation of state and experimental data shows high ability of this equation in predicting PρT properties of ionic liquids. The overall average absolute deviation percent of the calculated density of 15 ionic liquids obtained by this equation of state is 0.09%.

Generic van der Waals equation of state for polymers, modified free volume theory, and the self-diffusion coefficient of polymeric liquids

In this paper, a molecular theory of self-diffusion coefficient is developed for polymeric liquids (melts) on the basis of the integral equation theory for site–site pair correlation functions, the generic van der Waals equation of state, and the modified free volume theory of diffusion. The integral equations supply the pair correlation functions necessary for the generic van der Waals equation of state, which in turn makes it possible to calculate the self-diffusion coefficient on the basis of the modified free volume theory of diffusion. A random distribution is assumed for minimum free volumes for monomers along the chain in the melt. More specifically, a stretched exponential is taken for the distribution function. If the exponents of the distribution function for minimum free volumes for monomers are chosen suitably for linear polymer melts of N monomers, the N dependence of the self-diffusion coefficient is N − 1 for the small values of N , an exponent predicted by the Rouse theory, whereas in the range of 2.3 ≲ ln N ≲ 4.5 the N dependence smoothly crosses over to N − 2 , which is reminiscent of the exponent by the reptation theory. However, for ln N ≳ 4.5 the N dependence of the self-diffusion coefficient differs from N − 2 , but gives an N dependence, N − 2 − δ ( 0 < δ < 1 ) , consistent with experiment on polymer melts in the range. For polyethylene δ ≈ 0.48 for the parameters chosen for the stretched exponential. Because the stretched exponential function contains undetermined parameters, the N dependence of diffusion becomes semiempirical, but once the parameters are chosen such that the N dependence of D can be successfully given for a polymer melt, the temperature dependence of the self-diffusion coefficient can be well predicted in comparison with experiment. The theory is satisfactorily tested against experimental and simulation data on the temperature dependence of D for polyethylene and polystyrene melts.

Nonlinear diffusion in two-dimensional ordered porous media based on a free volume theory

A continuum nonlinear diffusion model is developed to describe molecular transport in ordered porous media. An existing generic van der Waals equation of state based free volume theory of binary diffusion coefficients is modified and introduced into the two-dimensional diffusion equation. The resulting diffusion equation is solved numerically with the alternating-direction fully implicit method under Neumann boundary conditions. Two types of pore structure symmetries are considered, hexagonal and cubic. The former is modeled as parallel channels while in case of the latter equal-sized channels are placed perpendicularly thus creating an interconnected network. First, general features of transport in both systems are explored, followed by the analysis of the impact of molecular properties on diffusion inside and out of the porous matrix. The influence of pore size on the diffusion-controlled release kinetics is assessed and the findings used to comment recent experimental studies of drug release profiles from ordered mesoporous silicates.

Excluded volume effect on entropy and clustering behaviors in supercooled liquids and glasses

We study the significant entropy changes in supercooling and glass transition by developing a two (H and L) state cluster model, where the H-state represents the clusters with the coordination number z being larger than the reference size z R and the L-state those with z < z R , respectively. We exploit the excluded volume effect on the clustering behaviors of various glass-formers and the temperature dependence (at fixed pressure of 1 atmosphere) of the entropy difference Δ s o between the two cluster states on the basis of z R = 1 + z R o [ exp ( − Δ s o / k B ) − 1 ] , where z R o = 1.8 as taken universally for all the glass-formers investigated. The temperature dependence of the average size 〈 z c ∗ 〉 of clusters with z ≥ z R is then determined for some representative glass-formers, such as GeO 2, 1-propanol, glycerol, propylene carbonate, o-terphenyl, and so forth, covering the entire fragility spectrum studied experimentally. As a result, the temperature dependence of 〈 z c ∗ 〉 is found to highly depend on the fragility index ϑ associated with the temperature dependence of the excluded volume B ∗ ( ϕ , T ) of which statistical mechanical representation is given by the generic van der Waals equation of state. With decreasing T toward the glass transition temperature T g , 〈 z c ∗ 〉 increases to attain universally about 13 at T g regardless of the fragility and molecular details of the glass-formers. We specifically analyze in depth the theoretical predictions for the cluster sizes and the experimental estimations for the length scales associated with temporary clusters of slow segments in supercooled glycerol as about 1.4±0.5 nm for glycerol at 10 K above T g . Given the large uncertainties associated with the experiment, the present theory appears to be in good agreement with the experiment.

Improved Method for the Self-Diffusion Coefficient in the Modified Free Volume Theory: Simple Fluids

In this paper, we further examine the modified free volume (MFV) theory of diffusion in an effort to improve the accuracy of self-diffusion coefficients calculated with the help of the generic van der Waals equation of state. It is shown that the self-diffusion coefficient of the Lennard-Jones (LJ) fluid can be improved significantly over the results obtained by the previously employed method. In the previous method, we have used the hard sphere self-diffusion coefficient for the pre-exponential factor in the MFV theory formula for the self-diffusion coefficient. We show in this paper that if the hard sphere self-diffusion coefficient used for the pre-exponential factor is replaced with the Chapman-Enskog formula for the LJ fluid the accuracy of the self-diffusion coefficient improves impressively in the liquid density for a wide range of temperature.

A Perturbation Method for the Ornstein−Zernike Equation and the Generic van der Waals Equation of State for a Square Well Potential Model

We calculate the generic van der Waals parameters A and B for a square well model by means of a perturbation theory. To calculate the pair distribution function or the cavity function necessary for the calculation of A and B, we have used the Percus-Yevick integral equation, which is put into an equivalent form by means of the Wiener-Hopf method. This latter method produces a pair of integral equations, which are solved by a perturbation method treating the Mayer function or the well width or the functions in the square well region exterior to the hard core as the perturbation. In the end, the Mayer function times the well width is identified as the perturbation parameter in the present method. In this sense, the present perturbation method is distinct from the existing thermodynamic perturbation theory, which expands the Helmholtz free energy in a perturbation series with the inverse temperature treated as an expansion parameter. The generic van der Waals parameters are explicitly calculated in analytic form as functions of reduced temperature and density. The van der Waals parameters are recovered from them in the limits of vanishing density and high temperature. The equation of state thus obtained is tested against Monte Carlo simulation results and found reliable, provided that the temperature is in the supercritical regime. By scaling the packing fraction with a temperature-dependent hard core, it is suggested to construct an equation of state for fluids with a temperature-dependent hard core that mimicks a soft core repulsive force on the basis of the equation of state derived for the square well model.

Voids, generic van der Waals equation of state, and transport coefficients of liquids

In this Perspective, we discuss the role of voids in transport processes in liquids and the manner in which the concept of voids enters the generic van der Waals equation of state and the modified free volume theory. The density fluctuation theory is then discussed and we show how the density fluctuation theory can be made a molecular theory with the help of the modified free volume theory and the generic van der Waals equation of state. The confluence of the aforementioned three theories makes it possible to calculate the transport coefficients of liquids by using the information on the equilibrium pair correlation function, which can be calculated either by an integral equation theory or Monte Carlo simulations. A number of relations between transport coefficients are also presented, which are derived on the basis of the density fluctuation theory. Since they can be used to obtain one transport coefficient from another they can be very useful in handling experimental and theoretical data. An application of the modified free volume theory to polymer melts is discussed as an example for a theory of transport properties of complex liquids.

Generalized excluded volume and the diffusivity and viscosity of supercooled liquids and glasses over the entire fragility spectrum

Transport properties of glass-formers near glass transition reflect the varying degrees of the sensitivity of the solid-like dynamics and structures with respect to temperature, depending on their fragility. Notably, however, most glasses resume Arrehenius transport behavior upon onset of vitrification. To address this phenomenon a theory of the self-diffusion coefficient and viscosity is developed on the basis of a model constructed for the generalized excluded volume of glass-formers described by the generic van der Waals equation of state. The molecular clustering behavior of a glass-former is exploited in terms of an order parameter that measures the concentration of glassy, clustered molecules, which is then related to the excluded volume. The formulas arrived therefrom are shown to excellently account for the self-diffusion coefficient and viscosity of various glass-formers over the entire fragility spectrum studied experimentally: e.g., GeO 2 , silica, ethanol, glycerol, diopside, propylene carbonate, o -terphenyl, tris-napthylbenzene, toluene, and so on. The excluded volume effect thus investigated is shown to essentially characterize the fragility of the glass-formers. The resulting theory not only predicts for fragile glass-formers to resume Arrehenius transport behavior upon the onset of the glass transition, but also explains a crossover between strong and fragile glass-formers in their diffusivity and viscosity profiles as vitrification sets in.

Self-diffusion Coefficient Model Based on Activation Energy and Free Volume

A new model for self-diffusion coefficients was proposed based on both the concepts of molecular free volume and activation energy. The unknown parameters of this model were clearly defined and compared with the Chapman-Enskog model. At the same time a new method for calculating activation energy was devised and applied to the new model. In addition, the free volume was defined by implementing the generic van der Waals equation of state, the radial distribution function of which was obtained by using the Morsali-Goharshadi empirical formula. Under the same conditions, the new model was better than the original free volume model.

Excluded volume in the generic van der Waals equation of state and the self-diffusion coefficient of the Lennard-Jones fluid

In the previous papers applying the generic van der Waals equation of state the mean excluded volume was defined with the contact diameter of particles at which the potential energy is equal to zero-the size parameter in the case of the Lennard-Jones potential. This parameter appears as the upper limit of the integral for the generic van der Waals parameter B (mean excluded volume divided by the density) in the generic van der Waals equation of state. Since the choice is not unique, in this paper we reexamine the manner of defining the upper limit and propose another choice for the upper limit. We also propose an interpretation of the free volume overlap factor alpha appearing in the free volume theory of diffusion and a method of estimating it in terms of the intermolecular potential energy only. It is shown that with the so-estimated free volume overlap factor and the new choice of the upper limit of the integral for B the self-diffusion coefficient in the modified free volume theory of diffusion not only acquires a better accuracy than before, but also becomes calculable in terms of only the intermolecular interaction potential without an adjustable parameter. We also assess some of effective diameters of molecules proposed in the literature for their ability to predict the self-diffusion coefficient within the framework of the modified free volume theory of diffusion.

Molecular theory of thermal conductivity of the Lennard-Jones fluid

In this paper the thermal conductivity of the Lennard-Jones fluid is calculated by applying the combination of the density-fluctuation theory, the modified free volume theory of diffusion, and the generic van der Waals equation of state. A Monte Carlo simulation method is used to compute the equilibrium pair-correlation function necessary for computing the mean free volume and the coefficient in the potential-energy and virial contributions to the thermal conductivity. The theoretical results are compared with our own molecular dynamics simulation results and with those reported in the literature. They agree in good accuracy over wide ranges of density and temperature examined in molecular dynamics simulations. Thus the combined theory represents a molecular theory of thermal conductivity of the Lennard-Jones fluid and by extension simple fluids, which enables us to compute the nonequilibrium quantity by means of the Monte Carlo simulations for the equilibrium pair-correlation function.

Relations between Transport Coefficients and Their Density and Temperature Dependence

Nonequilibrium statistical mechanics via density fluctuation theory predicts relations between the bulk and shear viscosity, thermal conductivity, and self-diffusion coefficient of a fluid. In this Feature Article, we discuss such relations holding for fluids over wide ranges of density and temperature experimentally studied in the laboratory. It is discussed how such relations can be used to successfully compute the density and temperature dependence on the basis of intermolecular interaction potential models with the help of the modified free volume theory and the generic van der Waals equation of state once the parameters in them are determined at a low density or at a subcritical temperature. Although some approximations have been made to derive them, they represent a reliable molecular theory of transport coefficients over the entire density and temperature ranges of fluids--namely, gases and liquids--a theory hitherto unavailable in the kinetic theory of liquids and dense gases.

Statistical-mechanical theory of rheology: Lennard-Jones fluids

The generalized Boltzmann equation for simple dense fluids gives rise to the stress tensor evolution equation as a constitutive equation of generalized hydrodynamics for fluids far removed from equilibrium. It is possible to derive a formula for the non-Newtonian shear viscosity of the simple fluid from the stress tensor evolution equation in a suitable flow configuration. The non-Newtonian viscosity formula derived is applied to calculate the non-Newtonian viscosity as a function of the shear rate by means of statistical mechanics in the case of the Lennard-Jones fluid. For that purpose we have used the density-fluctuation theory for the Newtonian viscosity, the modified free volume theory for the self-diffusion coefficient, and the generic van der Waals equation of state to compute the mean free volume appearing in the modified free volume theory. Monte Carlo simulations are used to calculate the pair-correlation function appearing in the generic van der Waals equation of state and shear viscosity formula. To validate the Newtonian viscosity formula obtained we first have examined the density and temperature dependences of the shear viscosity in both subcritical and supercritical regions and compared them with molecular-dynamic simulation results. With the Newtonian shear viscosity and thermodynamic quantities so computed we then have calculated the shear rate dependence of the non-Newtonian shear viscosity and compared it with molecular-dynamics simulation results. The non-Newtonian viscosity formula is a universal function of the product of reduced shear rate (gamma*) times reduced relaxation time (taue*) that is independent of the material parameters, suggesting a possibility of the existence of rheological corresponding states of reduced density, temperature, and shear rate. When the simulation data are reduced appropriately and plotted against taue*gamma* they are found clustered around the reduced (universal) non-Newtonian viscosity formula. Thus we now have a molecular theory of non-Newtonian shear viscosity for the Lennard-Jones fluid, which can be implemented with a Monte Carlo simulation method for the pair-correlation function.

Modified Free Volume Theory of Self-Diffusion and Molecular Theory of Shear Viscosity of Liquid Carbon Dioxide

In previous work on the density fluctuation theory of transport coefficients of liquids, it was necessary to use empirical self-diffusion coefficients to calculate the transport coefficients (e.g., shear viscosity of carbon dioxide). In this work, the necessity of empirical input of the self-diffusion coefficients in the calculation of shear viscosity is removed, and the theory is thus made a self-contained molecular theory of transport coefficients of liquids, albeit it contains an empirical parameter in the subcritical regime. The required self-diffusion coefficients of liquid carbon dioxide are calculated by using the modified free volume theory for which the generic van der Waals equation of state and Monte Carlo simulations are combined to accurately compute the mean free volume by means of statistical mechanics. They have been computed as a function of density along four different isotherms and isobars. A Lennard-Jones site-site interaction potential was used to model the molecular carbon dioxide interaction. The density and temperature dependence of the theoretical self-diffusion coefficients are shown to be in excellent agreement with experimental data when the minimum critical free volume is identified with the molecular volume. The self-diffusion coefficients thus computed are then used to compute the density and temperature dependence of the shear viscosity of liquid carbon dioxide by employing the density fluctuation theory formula for shear viscosity as reported in an earlier paper (J. Chem. Phys. 2000, 112, 7118). The theoretical shear viscosity is shown to be robust and yields excellent density and temperature dependence for carbon dioxide. The pair correlation function appearing in the theory has been computed by Monte Carlo simulations.

Generic van der Waals Equation of State, Modified Free Volume Theory of Diffusion, and Viscosity of Simple Liquids

The shear viscosity formula derived by the density fluctuation theory in previous papers is computed for argon, krypton, and methane by using the self-diffusion coefficients derived in the modified free volume theory with the help of the generic van der Waals equation of state. In the temperature regime near or above the critical temperature, the density dependence of the shear viscosity can be accounted for by ab initio calculations with the self-diffusion coefficients provided by the modified free volume theory if the minimum (critical) free volume is set equal to the molecular volume and the volume overlap parameter (alpha) is taken about unity in the expression for the self-diffusion coefficient. In the subcritical temperature regime, if the density fluctuation range parameter is chosen appropriately at a temperature, then the resulting expression for the shear viscosity can well account for its density and temperature dependence over the ranges of density and temperature experimentally studied. In the sense that once the density fluctuation range is fixed at a temperature, the theory can account for the experimental data at other subcritical temperatures on the basis of the intermolecular force only; the theory is predictive even in the subcritical regime of temperature. Theory is successfully tested in comparison with experimental data for self-diffusion coefficients and shear viscosity for argon, krypton, and methane.

Theory of the viscosity of supercooled liquids and the glass transition: fragile liquids.

A statistical mechanical theory is presented for viscosity of relatively low molecular weight organic liquids which are supercooled down to the glass transition temperature. In this theory a relation resembling the Stokes-Einstein relation between the viscosity and self-diffusion coefficient of supercooled liquids and an expression for the self-diffusion coefficient are augmented by a suitably constructed semiempirical generic van der Waals equation of state that makes it possible to calculate the free volume. The theory accounts in excellent accuracy for viscosities and self-diffusion coefficients of fragile liquids over the entire range of temperature experimentally investigated. According to the theory, vitrification occurs when the free volume available for translational molecular motion falls below a critical value.

Phenomenological Models for the Generic van der Waals Equation of State and Critical Parameters

In this paper, phenomenological models for the generic van der Waals equation of state are proposed for the subcritical regime of a simple fluid by assuming empirical forms for the generic van der Waals parameters A and B. The models that are assumed are tested against the critical parameters. It is shown that quadratic and cubic models for A and B are shown to give good critical parameters. The quadratic and cubic models are then employed to calculate critical isotherms of argon and methane to determine which model is superior. It is found that the quadratic model yields excellent critical isotherms in comparison with experimental data and is much superior to the cubic model in the experimental range of density. The model equation of state is also used to determine the spinodal curve, which is shown to be closely related to the nonanalytic behavior of A and B with respect to temperature.