Lennard-Jones Chains Research Articles

Linear, cyclic and branched Lennard-Jones chain quartic equation of state templates for associating fluid model development

Two new cubic equations of state (EOS's) for the pure Lennard-Jones (LJ) monomer fluid are constructed using molecular simulation data for compressibility factor Z versus packing fraction η isotherms. Simulation data for the LJ monomer and dimer pair correlation function at contact value derivatives are fit to extended (quartic) forms of these EOS's and used within the framework of Wertheim's first-order thermodynamic perturbation theory (TPT1) for both the monomer and dimer reference fluid to obtain ten new quartic in molar volume EOS's for straight, cyclic and branched LJ chain fluid isotherms. Predicted Z vs. η for the new equations is compared to simulation data and predictions from much more complex published models. Results show that these simpler equations are much easier to use, with minimal loss in accuracy. Finally, the LJ framework for individual cyclic chains is used in the Marshall–Chapman methodology for branched chains to derive two new LJ TPT1-D platforms and four new equations of state for “branched and fused rings”. The models presented in this paper serve as a foundation for the development of molecularly-based EOS's for real fluids with “soft” intermolecular and intramolecular potentials.

Conformation of a flexible polymer in explicit solvent: Accurate solvation potentials for Lennard-Jones chains.

The conformation of a polymer chain in solution is coupled to the local structure of the surrounding solvent and can undergo large changes in response to variations in solvent density and temperature. The many-body effects of solvent on the structure of an n-mer polymer chain can be formally mapped to an exact n-body solvation potential. Here, we use a pair decomposition of this n-body potential to construct a set of two-body potentials for a Lennard-Jones (LJ) polymer chain in explicit LJ solvent. The solvation potentials are built from numerically exact results for 5-mer chains in solvent combined with an approximate asymptotic expression for the solvation potential between sites that are distant along the chain backbone. These potentials map the many-body chain-in-solvent problem to a few-body single-chain problem and can be used to study a chain of arbitrary length, thereby dramatically reducing the computational complexity of the polymer chain-in-solvent problem. We have constructed solvation potentials at a large number of state points across the LJ solvent phase diagram including the vapor, liquid, and super-critical regions. We use these solvation potentials in single-chain Monte Carlo (MC) simulations with n ≤ 800 to determine the size, intramolecular structure, and scaling behavior of chains in solvent. To assess our results, we have carried out full chain-in-solvent MC simulations (with n ≤ 100) and find that our solvation potential approach is quantitatively accurate for a wide range of solvent conditions for these chain lengths.

Scaling of the dynamics of flexible Lennard-Jones chains: Effects of harmonic bonds.

The previous paper [A. A. Veldhorst et al., J. Chem. Phys. 141, 054904 (2014)] demonstrated that the isomorph theory explains the scaling properties of a liquid of flexible chains consisting of ten Lennard-Jones particles connected by rigid bonds. We here investigate the same model with harmonic bonds. The introduction of harmonic bonds almost completely destroys the correlations in the equilibrium fluctuations of the potential energy and the virial. According to the isomorph theory, if these correlations are strong a system has isomorphs, curves in the phase diagram along which structure, dynamics, and the excess entropy are invariant. The Lennard-Jones chain liquid with harmonic bonds does have curves in the phase diagram along which the structure and dynamics are invariant. The excess entropy is not invariant on these curves, which we refer to as "pseudoisomorphs." In particular, this means that Rosenfeld's excess-entropy scaling (the dynamics being a function of excess entropy only) does not apply for the Lennard-Jones chain with harmonic bonds.

Perturbation theory for multicomponent fluids based on structural properties of hard-sphere chain mixtures.

An analytical expression for the Laplace transform of the radial distribution function of a mixture of hard-sphere chains of arbitrary segment size and chain length is used to rigorously formulate the first-order Barker-Henderson perturbation theory for the contribution of the segment-segment dispersive interactions into thermodynamics of the Lennard-Jones chain mixtures. Based on this approximation, a simple variant of the statistical associating fluid theory is proposed and used to predict properties of several mixtures of chains of different lengths and segment sizes. The theory treats the dispersive interactions more rigorously than the conventional theories and provides means for more accurate description of dispersive interactions in the mixtures of highly asymmetric components.

A simple extrapolation of thermodynamic perturbation theory to infinite order.

Recent analyses of the third and fourth order perturbation contributions to the equations of state for square well spheres and Lennard-Jones chains show trends that persist across orders and molecular models. In particular, the ratio between orders (e.g., A3/A2, where A(i) is the ith order perturbation contribution) exhibits a peak when plotted with respect to density. The trend resembles a Gaussian curve with the peak near the critical density. This observation can form the basis for a simple recursion and extrapolation from the highest available order to infinite order. The resulting extrapolation is analytic and therefore cannot fully characterize the critical region, but it remarkably improves accuracy, especially for the binodal curve. Whereas a second order theory is typically accurate for the binodal at temperatures within 90% of the critical temperature, the extrapolated result is accurate to within 99% of the critical temperature. In addition to square well spheres and Lennard-Jones chains, we demonstrate how the method can be applied semi-empirically to the Perturbed Chain - Statistical Associating Fluid Theory (PC-SAFT).

An analytical equation of state for describing isotropic-nematic phase equilibria of Lennard-Jones chain fluids with variable degree of molecular flexibility.

We develop an equation of state (EoS) for describing isotropic-nematic (IN) phase equilibria of Lennard-Jones (LJ) chain fluids. The EoS is developed by applying a second order Barker-Henderson perturbation theory to a reference fluid of hard chain molecules. The chain molecules consist of tangentially bonded spherical segments and are allowed to be fully flexible, partially flexible (rod-coil), or rigid linear. The hard-chain reference contribution to the EoS is obtained from a Vega-Lago rescaled Onsager theory. For the description of the (attractive) dispersion interactions between molecules, we adopt a segment-segment approach. We show that the perturbation contribution for describing these interactions can be divided into an "isotropic" part, which depends only implicitly on orientational ordering of molecules (through density), and an "anisotropic" part, for which an explicit dependence on orientational ordering is included (through an expansion in the nematic order parameter). The perturbation theory is used to study the effect of chain length, molecular flexibility, and attractive interactions on IN phase equilibria of pure LJ chain fluids. Theoretical results for the IN phase equilibrium of rigid linear LJ 10-mers are compared to results obtained from Monte Carlo simulations in the isobaric-isothermal (NPT) ensemble, and an expanded formulation of the Gibbs-ensemble. Our results show that the anisotropic contribution to the dispersion attractions is irrelevant for LJ chain fluids. Using the isotropic (density-dependent) contribution only (i.e., using a zeroth order expansion of the attractive Helmholtz energy contribution in the nematic order parameter), excellent agreement between theory and simulations is observed. These results suggest that an EoS contribution for describing the attractive part of the dispersion interactions in real LCs can be obtained from conventional theoretical approaches designed for isotropic fluids, such as a Perturbed-Chain Statistical Associating Fluid Theory approach.

On the vapor-liquid equilibrium of attractive chain fluids with variable degree of molecular flexibility.

We study the isotropic (vapor and liquid) phase behavior of attractive chain fluids. Special emphasis is placed on the role of molecular flexibility, which is studied by means of a rod-coil model. Two new equations of state (EoSs) are developed for square-well- (SW) and Lennard-Jones (LJ) chain fluids. The EoSs are developed by applying the perturbation theory of Barker and Henderson (BH) to a reference fluid of hard chain molecules. The novelty of the approach is based on (1) the use of a recently developed hard-chain reference EoS that explicitly incorporates the effects of molecular flexibility, (2) the use of recent molecular simulation data for the radial distribution function of hard-chain fluids, and (3) a newly developed effective segment size, which effectively accounts for the soft repulsion between segments of LJ chains. It is shown that the effective segment size needs to be temperature-, density-, and chain-length dependent. To obtain a simplified analytical EoS, the perturbation terms are fitted by polynomials in density (SW and LJ), chain length (SW and LJ), and temperature (only for LJ). It is shown that the equations of state result in an accurate description of molecular simulation data for vapor-liquid equilibria (VLE) and isotherms of fully flexible SW- and LJ chain fluids and their mixtures. To evaluate the performance of the equations of state in describing the effects of molecular flexibility on VLE, we present new Monte Carlo simulation results for the VLE of rigid linear- and partially flexible SW- and LJ chain fluids. For SW chains, the developed EoS is in a good agreement with simulation results. For increased rigidity of the chains, both theory and simulations predict an increase of the VL density difference and a slight increase of the VL critical temperature. For LJ chains, the EoS proves incapable of reproducing part of these trends.

Cubic and quartic hard-sphere and Lennard-Jones chain equations of state as foundations for complex fluid modeling

Hard-sphere compressibility factor Z and hard-dimer radial distribution function at contact gHD(σ) are fit to simple packing fraction η expressions and are used in the dimer version of Wertheim’s perturbation theory to obtain five cubic equations of state (EOS’s) for athermal hard-sphere chain fluids. The predicted Zm versus η for various chain lengths m and the reduced second-virial coefficient versus m are compared with simulation data and the original noncubic dimer perturbation theory EOS’s. A cubic analog of the Boublik–Mansoori–Carnahan–Starling (BMCS) EOS successfully represents the unlike pair correlation function g12(σ) and Z versus η for binary hard-sphere mixtures of moderate size difference in the components. The new CTPT-D models are extended to calculate Zm versus η for binary homonuclear hard-sphere chain mixtures. The choice of a suitable EOS dispersion term to simultaneously represent single phase PVT and phase equilibria is briefly explored. A similar TPT approach was used to derive a quartic EOS for Lennard-Jones chain fluids, and Zm versus η isotherms are determined and compared with simulation results for chains up to 100 segments long. The new cubic and quartic EOS’s provide a simple foundation for developing equations of state for complex fluids including multipolar and association effects.

A unified approach to computation of solid and liquid free energy to revisit the solid-fluid equilibrium of Lennard-Jones chains.

Liquid free energies are computed by integration along a path from a reference system of known free energy, using a strong localization potential. A particular choice of localization pathway is introduced, convenient for use in molecular dynamics codes, and which achieves accurate results without the need to include the identity-swap or relocation Monte Carlo moves used in previous studies. Moreover, an adaptive timestep is introduced to attain the reference system. Furthermore, a center-of-mass correction that is different from previous studies and phase-independent is incorporated. The resulting scheme allows computation of both solid and liquid free energies with only minor differences in simulation protocol. This is used to re-visit solid-liquid equilibrium in a system of short semi-flexible Lennard-Jones chain molecules. The computed melting curve is demonstrated to be consistent with direct co-existence simulations and computed hysteresis loops, provided that an entropic term arising from unsampled solid states is included.

Chemical potentials and phase equilibria of Lennard-Jones chain fluids

Computer simulations (molecular dynamics) were performed for ensembles of flexible tangent Lennard-Jones chains consisting of n sites (n = 1, 2, 4, 8, and 16). From these simulations, the orthobaric liquid and vapour densities were calculated not only with the traditional method of simulating a liquid film in coexistence with vapour, but also using the rigorous thermodynamic condition of satisfying the chemical potential equality between the phases in equilibrium. The agreement with literature data, as far as such exist, is excellent.

Scaling of the dynamics of flexible Lennard-Jones chains.

The isomorph theory provides an explanation for the so-called power law density scaling which has been observed in many molecular and polymeric glass formers, both experimentally and in simulations. Power law density scaling (relaxation times and transport coefficients being functions of ρ(γ(S)), where ρ is density, T is temperature, and γ(S) is a material specific scaling exponent) is an approximation to a more general scaling predicted by the isomorph theory. Furthermore, the isomorph theory provides an explanation for Rosenfeld scaling (relaxation times and transport coefficients being functions of excess entropy) which has been observed in simulations of both molecular and polymeric systems. Doing molecular dynamics simulations of flexible Lennard-Jones chains (LJC) with rigid bonds, we here provide the first detailed test of the isomorph theory applied to flexible chain molecules. We confirm the existence of isomorphs, which are curves in the phase diagram along which the dynamics is invariant in the appropriate reduced units. This holds not only for the relaxation times but also for the full time dependence of the dynamics, including chain specific dynamics such as the end-to-end vector autocorrelation function and the relaxation of the Rouse modes. As predicted by the isomorph theory, jumps between different state points on the same isomorph happen instantaneously without any slow relaxation. Since the LJC is a simple coarse-grained model for alkanes and polymers, our results provide a possible explanation for why power-law density scaling is observed experimentally in alkanes and many polymeric systems. The theory provides an independent method of determining the scaling exponent, which is usually treated as an empirical scaling parameter.

Equilibrium, interfacial and transport properties of n-alkanes: Towards the simplest coarse grained molecular model

In this work, using molecular dynamics simulations, we have investigated how simple could be a coarse grained molecular model yielding simultaneously equilibrium densities, surface tensions and transport properties of some n-alkanes (from methane to n-decane) along the vapour–liquid equilibrium curve. For that purpose, as an initial model, the fully flexible Lennard–Jones Chain (3 “molecular” parameters) model has been chosen. Using this simple molecular model, good results have been obtained on equilibrium properties of all tested n-alkanes despite a systematic slight overestimation of critical temperatures, critical pressures and surface tensions. In addition, concerning transport properties, good results have been obtained for methane and n-butane except for thermal conductivity in the gas state. For n-heptane and n-decane it has been found that thermal conductivity is systematically underestimated while viscosity is well estimated except at low temperatures. Concerning thermal conductivity, this misevaluation can be corrected if the zero-density thermal conductivity is known. Concerning shear viscosity, it is found that an additional “rigidity” fourth parameter is required to improve the results when dealing with the longest chains at the lowest temperatures.

An examination of the excess thermodynamic properties of flexible molecules from a molecular modelling perspective

Excess thermodynamic properties provide fundamental information on the intermolecular interactions in multicomponent fluid mixtures. However, very often only phase equilibria is used to test the accuracy of any theory or molecular model describing the solution thermodynamic behaviour. Although this approach is valid and often sufficient, some situations require stronger tests to determine whether a model and/or theory can provide a realistic description of a particular system. Excess properties estimation is a valuable test for this purpose because excess properties are very sensitive to the molecular model details. The goal of this work is to show how simple models, with a reduced number of molecular parameters, are able to predict accurately excess thermodynamic properties. We concentrate on binary mixtures formed by different models of flexible molecules. In particular, we focus on two general models: the fully-flexible tangentially bonded Lennard–Jones chains model described with the SAFT VR [J. Chem. Phys. 106 (1997) 4168] approach, and the united-atom model combined with Monte Carlo simulation, in both cases applied to the description of short chain alkane mixtures (from methane to propane). While the first model does not consider intramolecular interactions and the bond length equals the monomer size, the second one incorporates a more realistic description. Although the molecular models are relatively simple, they are able to describe most of the microscopic features of real chainlike molecules. The predictions obtained from SAFT-VR and simulation are compared with available experimental data, and the agreement is good for a wide range of thermodynamic conditions. In addition to the vapour–liquid equilibria, both theory and simulation are able to characterize the most relevant features of two important excess thermodynamic properties, excess volume and enthalpy.

Do Transport Properties of Entangled Linear Polymers Scale with Excess Entropy?

We analyze the range of validity of the empirical excess entropy scalings proposed independently by Rosenfeld and Dzugutov for the viscosity and the diffusion coefficient of monodisperse Lennard-Jones chains. They are long enough to be considered as entangled. Thus, the influence of entanglements on entropy scalings can be quantified. We make use of thermodynamic integration to determine the exact excess entropy. Different ways of approximating the excess entropy either based on conformational information or by employing the self-associating fluid theory (SAFT) are explored. The correlations between the various excess entropy estimates are investigated in detail. The conformational route appears to be the most suitable way for the studied Lennard-Jones systems. Its prediction is in qualitative agreement with those obtained by thermodynamic integration, which is computationally more demanding. The SAFT results are not always consistent with the simulation data. The main qualitative feature of both the Rosenfeld and Dzugutov scaling is the coexistence of two linear regimes in the correlation of different entropy expressions. The sharpness of the transition regime depends on the way the excess entropy is approximated. The Dzugutov scaling for the viscosity seems to have certain advantages in comparison to the Rosenfeld one. The influence of the chain length on the parameters appearing in the excess entropy scaling relationships is examined. Despite the pronounced chain length dependence of the scaling parameters, a master curve relating the diffusion coefficient to the excess entropy is derived when performing a suitable renormalization. The relation between the scaling parameters of the viscosity and the diffusion coefficient are—with one exception—in line with the predictions of the Stokes–Einstein law.

Free-Volume Theory Coupled with Soft-SAFT for Viscosity Calculations: Comparison with Molecular Simulation and Experimental Data

The evaluation of phase equilibria and solubility properties through theoretical approaches is a well-known field, where a significant amount of models are able to describe them with a good degree of accuracy. However, the simultaneous calculation of transport properties together with thermodynamic phase properties still remains a challenge, due to the difficulties in describing the behavior of properties like the viscosity of fluids with the same approach. In this work, the free-volume theory (FVT) has been coupled with the soft-SAFT equation for the first time to extend the capabilities of the equation to the calculation of transport properties. The theory has been first tested using simulation data of the viscosity of the Lennard-Jones (LJ) fluid and LJ chains over a wide range of temperature and pressure. Good agreement has been found at all chain lengths, except for some deviations at near-zero density values. Several trends of the viscosity parameters with the length of the chain are identified, allowing the prediction of other chain fluids. Finally, the new equation has been applied to the n-alkanes family, where viscosity is a key property, and results are compared with experimental data. The three viscosity parameters were fitted to viscosity data of the pure fluid at several isotherms or isobars, whereas the density and pressure (or temperature) were taken from the soft-SAFT output. Again, the effect of these parameters on the viscosity has been investigated and compared with results obtained for the LJ chains and with previous work of other authors. The new equation performs very well in all cases, with a global average absolute deviation of 2.12% and shows predictive capabilities for heavier compounds. This empowers soft-SAFT with new capabilities, allowing the equation to calculate phase, interfacial, and transport properties with the same model and degree of accuracy.

Calculation of the relaxation properties of a dilute gas consisting of Lennard–Jones chains

The relaxation properties in the dilute-gas limit have been calculated by the classical trajectory (CT) method for a gas consisting of chain-like molecules that are rigid and interact through site–site Lennard–Jones 12–6 potentials. Results are reported for volume viscosity ηV, rotational collision number ζrot and the ratio of the rotational to self-diffusion coefficient Drot/D. The results indicate that the volume viscosity increases with temperature and decreases with chain length. The rotational relaxation of chains is efficient, as it takes of the order of 1.75–2.7 collisions to attain equilibrium. The rotational collision number is only weakly temperature dependent.

Three new branched chain equations of state based on Wertheim's perturbation theory

In this work, we present three new branched chain equations of state (EOS) based on Wertheim's perturbation theory. The first represents a slightly approximate general branched chain solution of Wertheim's second order perturbation theory (TPT2) for athermal hard chains, and the second represents the extension of first order perturbation theory with a dimer reference fluid (TPT1-D) to branched athermal hard chain molecules. Each athermal branched chain EOS was shown to give improved results over their linear counterparts when compared to simulation data for branched chain molecules with the branched TPT1-D EOS being the most accurate. Further, it is shown that the branched TPT1-D EOS can be extended to a Lennard-Jones dimer reference system to obtain an equation of state for branched Lennard-Jones chains. The theory is shown to accurately predict the change in phase diagram and vapor pressure which results from branching as compared to experimental data for n-octane and corresponding branched isomers.

Calculation of the transport properties of a dilute gas consisting of Lennard-Jones chains

The transport properties in the dilute gas limit have been calculated by the classical-trajectory method for a gas consisting of chain-like molecules. The molecules were modelled as rigid chains consisting of spherical segments that interact through a combination of site-site Lennard-Jones 12-6 potentials. Results are reported for shear viscosity, self-diffusion, and thermal conductivity for chains consisting of 1, 2, 3, 4, 5, 6, 7, 8, 10, 13, and 16 segments in the reduced temperature range of 0.3-50. The results indicate that the transport properties increase with temperature and decrease with chain length. At high temperatures the dependence of the transport properties is governed effectively by the repulsive part of the potential. No simple scaling with chain length has been observed. The higher order correction factors are larger than observed for real molecules so far, reaching asymptotic values of 1.019-1.033 and 1.060-1.072 for viscosity and thermal conductivity, respectively. The dominant contribution comes from the angular momentum coupling. The agreement with molecular dynamics calculations for viscosity is within the estimated accuracy of the two methods for shorter chains. However, for longer chains differences of up to 7% are observed.

Low density shear viscosity of Lennard-Jones chains of variable rigidities

The zero-density shear viscosity of different types of short Lennard-Jones chains, up to the hexa-decamer, has been evaluated using a non-equilibrium molecular dynamics scheme. Simulations have been performed on chains of variable rigidities going from the fully flexible to the fully rigid chains. Very interestingly, it is found that there exists a universal relation (a power law) between the zero-density viscosity of the Lennard-Jones chains and their radius of gyration whatever the rigidity of the chain and for all tested temperatures (ranging from 2.5 to 6 in reduced units). Furthermore, for the studied range of temperature, it is shown that the zero-density viscosity of both fully flexible chains and fully rigid chains models can be obtained with an accuracy of a few percents knowing only the dimer viscosity and the length of the chain.

A test of systematic coarse-graining of molecular dynamics simulations: Thermodynamic properties

Coarse-graining (CG) techniques have recently attracted great interest for providing descriptions at a mesoscopic level of resolution that preserve fluid thermodynamic and transport behaviors with a reduced number of degrees of freedom and hence less computational effort. One fundamental question arises: how well and to what extent can a "bottom-up" developed mesoscale model recover the physical properties of a molecular scale system? To answer this question, we explore systematically the properties of a CG model that is developed to represent an intermediate mesoscale model between the atomistic and continuum scales. This CG model aims to reduce the computational cost relative to a full atomistic simulation, and we assess to what extent it is possible to preserve both the thermodynamic and transport properties of an underlying reference all-atom Lennard-Jones (LJ) system. In this paper, only the thermodynamic properties are considered in detail. The transport properties will be examined in subsequent work. To coarse-grain, we first use the iterative Boltzmann inversion (IBI) to determine a CG potential for a (1-φ)N mesoscale particle system, where φ is the degree of coarse-graining, so as to reproduce the radial distribution function (RDF) of an N atomic particle system. Even though the uniqueness theorem guarantees a one to one relationship between the RDF and an effective pairwise potential, we find that RDFs are insensitive to the long-range part of the IBI-determined potentials, which provides some significant flexibility in further matching other properties. We then propose a reformulation of IBI as a robust minimization procedure that enables simultaneous matching of the RDF and the fluid pressure. We find that this new method mainly changes the attractive tail region of the CG potentials, and it improves the isothermal compressibility relative to pure IBI. We also find that there are optimal interaction cutoff lengths for the CG system, as a function of φ, that are required to attain an adequate potential while maintaining computational speedup. To demonstrate the universality of the method, we test a range of state points for the LJ liquid as well as several LJ chain fluids.