Hard Chain Research Articles

Chemical potential of the two dimensional fluid and solid phases

The statistical associating fluid theory (SAFT) for hard sphere chain fluids has been extended to the two-dimensional hard disk chain in fluid and solid phases. For fluids, SAFT dimer has been employed to determine the equation of state (EOS) and chemical potential of the hard disk chain fluids for various chain lengths. For solids, the SAFT also known as Wertheim's first-order perturbation theory has been employed to determine the equation of state and chemical potential of the two dimensional hard homonuclear the dumbbell-shaped Herringbone (HB) crystal with a high anisotropy parameter (ℓ*=0.924). The fluid–solid transition for dumbbell shape (ℓ*=0.924) molecule has also been determined.

Algorithmic Generation of Freely Jointed Hard Sphere Chains and Properties of Inertial Tensors

A statistical algorithm, capable of generating a large number of freely jointed hard sphere chains, is presented. This is the first of a series of algorithms being developed to model unfolded proteins by different modes of hard sphere chains. The aim of these studies is to systematically investigate the effects of different factors, such as atomic radii, bond angles, torsion angles, chain length, etc., on the conformation of unfolded proteins and other random polymers. As continuous models, various types of hard sphere chains enable one to isolate the aforementioned factors one at a time for investigation and thus are advantageous over discrete lattice models. In particular, the freely jointed hard sphere chain model allows one to evaluate the excluded volume effect. As a first step in this endeavor, the average determinant D(N,r) and the average trace T(N, r) of the inertial tensor A of the random chains were calculated at various sphere radii r and chain lengths N. It is found that both the average determinant D(N,r) and the average trace T(N,r) scale linearly with chain length N. after logarithmic transformation. However, the critical exponent of D(N,r) increases with r faster than that of T(N,r) as a result of the non-commutativity between the det operator and the average operator < >. The significance of the algorithm and the results obtained on understanding random polypeptide chains are discussed.

Statistical associating Fluid Dimer Theory

Three modified versions of statistical associating fluid theory known as SAFT-D1, SAFT-D2 and SAFT-T for hard sphere chain molecules are used to test the approximation made in the derivation of the statistical associating fluid dimer theory (SAFT-D). The SAFT-D1 model accounts for the chain length independent site–site correlation function at contact, while SAFT-D2 and SAFT-T models account for chain length dependent site–site correlation function at contact. The residual chemical potentials are calculated for hard chain fluids containing chains of length m = 3, 4, 6, 8, 12, 16 and 32 mers and compared with the results obtained by configurational bias Monte Carlo (MC) method at various densities. We find that the approximation used in SAFT-D1 theory leads to an over prediction of residual chemical potential and its magnitude increases with increasing chain length.

Motional heterogeneity and phase separation of functionalized polyester polyurethanes

The polyester polyurethanes, PU based on isophoronediisocyanate, polycaprolactone, and 1,4-butanediol with different amounts of functional groups introduced into the hard segments via second chain extender, 2,2′-bis-(hydroxymethyl) propionic acid, were investigated by electron spin resonance, ESR, spin label method, wide-angle X-ray diffraction, WAXD, optical microscopy and differential scanning calorimetry, DSC. The objective of this study is to clarify the effect of functional groups on the motional heterogeneity, microphase separation and crystallisation of the polyurethanes. The concentration of carboxylic groups varied from 0 to 0.45 mmol g −1. The temperature-dependent ESR spectra of spin labelled PU hard segments chain ends with stable nitroxide radical 2,2,6,6-tetramethyl-4-aminopiperidin-1-yloxyl are sensitive to the amount of functional groups attached to the hard segments. Composite ESR spectra of functionalized PU, with fast and slow component, suggest that PU hard segments are partitioned in two motionally different environments. According to the ratio of fast and slow component motional heterogeneity increases with an increase of functional groups up to 0.35 mmol g −1 and above this concentration slow component decreases indicating higher degree of phase mixing and stronger effect of soft segments. Polarized micrographs and the extent of ordering from WAXD measurements reveal the changes of phase morphology with the carboxylic groups content in a similar way as shown from the motional behaviour of spin label on the segmental level. The degree of crystallinity and the separation of spherulitic rings are decreasing above a certain concentration of functional groups. The effect of functional groups in PU on the hard and soft segment mixing is discussed in terms of additional noncovalent interactions and chain structure which at critical level of interactions lead to a formation of more open hard segment structure accessible to interaction with the soft segment.

Thermodynamic perturbation theory for fused sphere hard chain fluids using nonadditive interactions.

A model is developed for the equation of state of fused chains based on Wertheim thermodynamic perturbation theory and nonadditive size interactions. The model also assumes that the structure (represented by the radial distribution function) of the fused chain fluid is the same as that of the touching hard sphere chain fluid. The model is completely based on spherical additive and nonadditive size interactions. The model has the advantage of offering good agreement with simulation data while at the same time being independent of fitted parameters. The model is most accurate for short chains, small values of Delta (slightly fused spheres) and at intermediate (liquidlike) densities.

Solid–fluid and solid–solid phase equilibrium in a model of n-alkane mixtures

Solid-fluid and solid-solid phase equilibrium for binary mixtures of hard sphere chains modeling n-hexane, n-heptane, and n-octane has been calculated using Monte Carlo computer simulations. Thermodynamic integration was used to calculate the Gibbs free energy and chemical potentials in the solid and fluid phases from pure component reference values. A multiple stage free energy perturbation method was used to calculate the composition derivative of the Gibbs free energy. Equation of state and free energy data for the fluid phase indicate ideal solution behavior. Nonideality is much more significant in the solid phase with only partial solubility of shorter chains in the longer chains and essentially no solubility at the other end of the composition range. The miscibility decreases with increasing chain length difference between the components. For the model of n-hexane/n-octane mixtures solid--solid phase separation has been observed directly in some of the simulations, with the components segregating between the layers of the solid structure. The behavior is similar to that seen in some binary n-alkane mixtures with longer chain lengths but comparable chain length ratios between the components. Such phase separation, although indicated thermodynamically, is not seen directly in the simulations of the n-heptane/n-octane mixture due to the difference in the pure component crystal structures.

Theory of the Crystallization of Hard Polymeric Chains in an Orienting Field

We apply a recently proposed density functional theory (DFT) for hard polymeric chains to study the influence of a quadrupolar orienting field on the freezing of model polymers. We find that a “disorienting” field, mimicking elongational flow, promotes crystallization, while an “orienting” field, which corresponds to longitudinal shear flow, destabilizes the crystal phase. We also investigate the relative stability of two close-packed crystal lattice types, face-centered cubic and hexagonal close-packed. Geometrical effects, such as the orientation of the field with respect to the main axes of the crystal, play an important role in stabilizing one crystal type over another.

Application of an improved simplified perturbed hard chain theory equation of state for pure compounds and binary asymmetric mixtures

The simplified perturbed hard chain theory equation of state is modified using recently developed repulsive and attractive terms. Both terms meet the low density and close-packed density boundary conditions and are in reasonable agreement with molecular simulation data. The modified equation of state accurately predicts pure component properties including saturated vapor volume, liquid density, vapor pressure and enthalpy of vaporization. Compared with the original equation, the modified one predicts physical properties more accurately. However, the improvement of predicting enthalpy of vaporization is marginal. The two equations are tested for predicting the phase behavior of binary asymmetric mixtures. It is shown that the difference in predicting the phase behavior is not appreciable.

Investigation of the Microstructure of Micelles Formed by Hard-Sphere Chains Interacting via Size Nonadditivity by Discontinuous Molecular Dynamics Simulation

Micelle formation by short nonadditive hard surfactant chains was investigated at different size ratios, reduced densities, and nonadditivity parameters using molecular dynamics simulation. It was found that spherical, cylindrical, lamellar, and reverse micelles can form in systems with different head, tail, and solvent characteristics. Hard-core surfactant chains composed of a head segment and three tail segments were simulated in a solvent of hard spheres. The formation of micelles was found to be a strong function of the packing fraction and nonadditivity parameter. Micelles were more stable at higher densities and larger nonadditivity parameters. At lower densities, micelles tended to break into small, dynamic globules.

Influence of monomer sequence on microstructure of nonadditive hard chain copolymers: Simulation and equation of state

Influence of copolymer structure on self-assembly of the copolymer was investigated by introducing nonadditive size interactions among different type segments in copolymers composed of touching hard spheres. The copolymer structures investigated are diblock, alternating, multiblock, and random. The copolymers were simulated using discontinuous molecular dynamics simulation. Equations of state of the different model copolymers were developed using the TPT1 theory. Diblock copolymers were found to self assemble readily at high values of the nonadditivity parameter Δ and at high densities. Multiblock and random copolymers self assembled to a smaller extent and the alternating copolymer self-assembled only at very high Δ and densities. The TPT1 equation of state provided good prediction of the compressibilities of the different copolymer systems at negative and small positive values of Δ. Deviations between the model and simulation results were observed at high Δ and high densities. Deviations were associated with self-assembly of the copolymers.

Fluid–solid equilibria of flexible and linear rigid tangent chains from Wertheim’s thermodynamic perturbation theory

An extension of Wertheim’s first-order thermodynamic perturbation theory is proposed to describe the global phase behavior of linear rigid tangent hard sphere chains. The extension is based on a scaling proposed recently by Vega and McBride [Phys. Rev. E 65, 052501 (2002)] for the equation of state of linear chains in the solid phase. We have used the Einstein-crystal methodology, the Rahman–Parrinello technique, and the thermodynamic integration method for calculating the free energy and equation of state of linear rigid hard sphere chains with different chain lengths, including the solid–fluid phase equilibria. Agreement between the simulation data and theoretical predictions is excellent in all cases. Once it is confirmed that the proposed theory can be used to describe correctly the equation of state, free energy, and solid–fluid phase transitions of linear rigid molecules, a simple mean-field approximation at the level of van der Waals is included to account for segment–segment attractive interactions. The approach is used to determine the global phase behavior of fully flexible and linear rigid chains of varying chain lengths. The main effect of increasing the chain length in the case of linear rigid chains is to decrease the fluid densities at freezing, so that the triple-point temperatures increase. As a consequence, the range of temperatures where vapor–liquid equilibria exist decreases considerably with chain length. This behavior is a direct result of the stabilization of the solid phase with respect to the liquid phase as the chain length is increased. The vapor–liquid equilibria are seen to disappear for linear rigid chains formed by more than 11 hard sphere segments that interact through an attractive van der Waals mean-field contribution; in other words, long linear rigid chains exhibit solid–vapor phase behavior only. In the case of flexible chains, the fluid–solid equilibrium is hardly affected by the chain length, so that the triple-point temperature reaches quickly an asymptotic value. In contrast to linear rigid chains, flexible chains present quite a broad range of temperatures where vapor–liquid equilibria exist. Although the vapor–liquid equilibria of flexible and linear rigid chain molecules are similar, the differences in the type of stable solid they form and, more importantly, the differences in the scaling of thermodynamic properties with chain length bring dramatic differences to the appearance of their phase diagrams.

Conformational behavior of a single polymer chain confined by a two-dimensional harmonic potential in good solvents

Monte Carlo simulations have been conducted to investigate the conformational behavior of a flexible chain polymer confined to a two-dimensional harmonic potential. The polymer molecule is modeled as a tangent hard sphere chain, and the two-dimensional harmonic potential is chosen to mimic nonrigid cylindrical pores. The simulations show that as field strength is increased, the mean chain dimension decreases first and then increases again after passing a minimum due to anisotropic deformation. A modified Flory-type theory is utilized to derive the power laws for the chain deformation against the strength of the applied field in different directions. These power laws agree with the simulations at strong fields when the confined polymer molecule exhibits a rodlike conformation. Meanwhile, a simple model, consisting of a dimer in a two-dimensional harmonic potential, is solved to elucidate the alignment of chain segments in very strong applied potentials. From this model, the crossover regime of a tangent hard sphere chain from the rodlike chain to a totally stretched chain at limiting strong fields is identified. Furthermore, a first-order perturbation theory is employed to interpolate the mean chain size for different field strengths. The field strength corresponding to the minimum mean chain size decreases as chain length is increased, consistent with the prediction of the modified Flory-type theory. These studies provide physical insights into the conformational behavior of a flexible polymer chain in nonspherical two-dimensional harmonic potentials.

Theoretical consideration on phase behaviors of poly(ethylene oxide-block-propylene oxide)/LiCF 3SO 3 systems in lithium battery

A new thermodynamic model is developed based on the extended perturbed hard sphere chain (PHSC) model and melting point depression theory to describe the phase behaviors of copolymer electrolyte/salt systems. The phase behaviors of poly(ethylene oxide-block-propylene oxide)/LiCF 3SO 3 systems are investigated by thermo-optical analysis (TOA) technique. Quantitative descriptions according to the proposed model are in good agreement with experimental data. The obtained results show that monomer ratio and sequence type of copolymers play a great role in determining eutectic points of the given systems.

Virial coefficients and equation of state of hard chain molecules

The second, third, and fourth virial coefficients of hard chain molecules with number of segments up to 10 (up to 7 in the case of the fourth one) and the reduced distances L*=1 and 0.5 were determined numerically. For fused hard sphere (FHS) molecules the enlarged fused hard sphere model is introduced which forms a connection to the hard convex geometry and makes it possible to determine the virial coefficients of FHS molecules from the expressions derived for hard convex bodies. Our numerical values of the virial coefficients together with data from literature are used to test the proposed method and to compare its results with values from Wertheim’s theory, from its modified version and from the generalized Flory dimer approach. It is found that prediction of the second virial coefficient is very accurate (for L=0.5 the maximum deviation amounts 0.2 percent) and that our values of the third virial coefficient for higher members of the FHS family are superior to those from other considered methods. The model is successfully used to predict geometric characteristics of the hard models of n-butane conformers and to estimate their second virial coefficients. The equation of state of hard prolate spherocylinders in which the nonsphericity parameters of the enlarged FHS model are considered is used to predict the P-V-T behavior of hard dumbbells, hard triatomics, mixture of hard dumbells of different site–site distances, and n-butane taken as a mixture of conformers. In all the cases prediction within error bars is obtained.

Understanding liquid-liquid immiscibility and LCST behaviour in polymer solutions with a Wertheim TPT1 description

The aim of the work presented in this paper is to help in the understanding of the lower critical solution temperature (LCST) fluid phase behaviour exhibited by polymer solutions. It is well recognized that the LCST in polymer solutions is a consequence of density (compressibility) effects; the solvent is much more compressible than the polymer and the increasing difference in compressibility when the temperature is increased leads to a negative volume of mixing. The separate roles that the repulsive and attractive intermolecular interactions play in this regard are less well understood. In this study we use the Wertheim first-order thermodynamic perturbation theory (TPT1) [Wertheim, M. S., 1987, J. chem. Phys., 87, 7323; Chapman, W. G., Jackson, G., and Gubbins, K. E., 1988, Molec. Phys., 65, 1057] to describe the phase equilibria of model polymer solutions of hard spheres and hard-sphere chains where the diameter of the solvent and the polymeric segments are the same (symmetrical system). The thermodynamic functions (volume, enthalpy, entropy and Gibbs function) of mixing are determined to assess the possibility of a demixing instability in such a system. No fluid-fluid phase separation is found for the purely repulsive (athermal) system, regardless of the chain length of the polymer. The role of the attractive interactions is then investigated by incorporating attractive interactions at the mean-field level; the simplest system with equivalent (symmetric) solvent-solvent, solvent-polymer segment, and polymer segment-polymer segment interaction energies is examined. The attractive interactions are found to be essential in describing the liquid-liquid phase separation; LCST behaviour is found for mixtures with ‘polymer’ chains of seven segments or more. In this case we show that the phase behaviour is driven by an unfavourable (negative) entropy of mixing due to an increase in the density of the solvent on addition of small amounts of polymer. We also determine the thermodynamic properties of mixing for a system of spherical molecules of the same size with directional interactions that give rise to LCST and closed-loop behaviour. As expected the mechanism for phase separation in such systems is very different to that in polymer solutions.

Virial coefficients from equation of state

This paper presents the theoretical prediction of second, third and fourth virial coefficients of linear hard chain molecules employing equations of state. The equations of state are determined through statistical associating fluid theory (SAFT). Two improvements of SAFT theory as SAFT-Dimer and SAFT-Trimer are employed to determine the virial coefficients. The SAFT-Trimer model predicts the virial coefficients much closer to the pearl-necklace model in comparison to the SAFT-Dimer model, concluding thereby that the SAFT-Trimer model seems more flexible in comparison to that of the dimer model.

Monte Carlo simulations of conformations of short chains near a cylindrical rod

Off-lattice Monte Carlo simulations are employed to study the behavior of short chain molecules near a long cylindrical rod. The molecules are modeled by freely jointed hard sphere chains. The emphasis is placed on the chain and bead distributions as well as the location dependence of the chain conformation and anisotropy. The simulation results show that at a given low chain concentration, the surface curvature of the rod gives rise to a slight increase in the bead concentration within the depletion layer. As the separation distance from the rod decreases, the chains first contract slightly and exhibit weak anisotropy in favor of an orientation parallel to the rod axis. With further increase in the closeness, to the contrary, the chains elongate and adopt a conformation that tends to wrap around the rod. This intriguing behavior is attributed primarily to the effect of the cylindrical surface.

Integral equation theories for monodisperse and polydisperse sticky hard sphere chain fluid: Thermodynamic and structural properties in the polymer Percus–Yevick and ideal chain approximations

The thermodynamic and structural properties of a freely jointed homonuclear sticky hard sphere chain fluid are studied using one-component and multi-component multi-density Ornstein–Zernike integral equation theories. In this formalism, a polydisperse chain fluid is modeled as a one-component system of associating sticky hard spheres with finite association strength, while a monodisperse system is modeled as an equal molar m-component mixture of associating sticky hard spheres with infinite association strength in the complete association limit. General analytical solutions to both models are obtained within the polymer Percus–Yevick and ideal chain approximations. Explicit analytical expressions for the contact values of correlation functions are obtained. The coordination number around a sticky hard sphere is calculated and its relationship with the contact value of the correlation function is discussed. Both intermolecular and intramolecular correlation functions beyond the hard core region are calculated numerically. Radial distribution functions of monodisperse dimer and tetramer are compared with those of polydisperse chains (with mean chain lengths of 2 and 4), and it is found that significant discrepancy exists at low density. This disparity, however, diminishes as chain density increases. The Helmholtz energy and pressure of monodisperse homonuclear chains are obtained via the energy route. The critical temperature, critical density, and phase coexistence of the fluid are also obtained.

Theory of melting point depression for SPE/salt systems; applicability of the extended PHSC model

We derive a new melting point depression theory based on the extended perturbed hard sphere chain (PHSC) model to interpret phase behaviors of solid polymer electrolyte systems with various salts. Quantitative descriptions according to the proposed model are in good agreement with experimental data for the given systems. The obtained results show that the salt weight fraction and the size of salt molecule play a great role in determining eutectic points of the given systems.

Density functional theory for the elastic moduli of a model polymeric solid

We apply a recently developed density functional theory for freely hinged, hard polymeric chains to calculate the elastic moduli of an idealized polymeric solid lacking long-range bond order. We find that for such a model packing effects dominate the elastic behavior of the polymeric solid in a similar way as is the case in the hard-sphere crystal, which we reexamine. Our calculations show that the elastic stiffness of the model polymeric solid is essentially determined by how far one is removed from its melting point. The main role of the chain connectivity is to destabilize the solid relative to the equivalent solid of hard monomers. Comparison of our results with experimental data on semicrystalline polymers shows order-of-magnitude agreement.