Intramolecular Correlation Functions Research Articles

Molecular dynamics simulation of the structure of an ion-conducting PEO-based solid polymer electrolyte

Proton-conducting solid polymer electrolyte containing poly(ethylene oxide) sulfonic acid anions (PEO) cations and 35 wt% of water is constructed by atomistic molecular modelling. The structure of the PEO sulfonic acid anions and PEO was studied by calculating the dihedral angle distributions, intramolecular pair correlation functions between the carbon atoms and ether oxygen atoms and by performing the conformational triads population analysis and dimension analysis. The results were compared with the results obtained for similar non-conducting systems and for the system containing only one PEO sulfonic acid anion in water. Some differences were found in the structures. However, the differences in the structures of the polymers were not able to explain the conductivity behaviour of the systems. Instantaneous coordination of cations was studied to get a detailed view of the local environment of the cations. The local environment of the cations shows many variations in the studied systems. Small microphase separation in the conducting system was found.

Integral equations for polymers in quenched random media

Integral equations are derived for the self-consistent evaluation of intramolecular and intermolecular correlation functions of polymers in quenched random media. The theory uses the replica trick to average over the quenched disorder using the Given–Stell procedure. The equations derived by Chandler in a heuristic fashion are recovered under certain approximations.

Molecular dynamics simulation of the structure of PEO based solid polymer electrolytes

A water-free solid polymer electrolyte containing poly(ethyle oxide) (PEO) and poly(ethyle oxide) with sulfonic acid dianion end groups (PEO sulfonic acid dianion) is constructed using periodic boundary conditions by atomistic molecular modelling. The vibrational frequencies are calculated and found to be in agreement with the measured spectra. The structure of the PEO sulfonic acid dianion is studied by calculating the dihedral angle distributions and the intramolecular pair correlation functions between the carbon atoms and the ether oxygen atoms. The dimensional analysis is made. Results concerning the PEO sulfonic acid dianion in the water-free system are compared to the results obtained in the water containing systems and in vacuum. The structure of the PEO sulfonic acid dianion in water-free system is found to be rather similar to its structure in water and PEO containing system but different from its structure in vacuum.

Entropy-driven polymer collapse: Application of the hybrid MC/RISM method to the study of conformational transitions in macromolecules interacting with hard colloidal particles

Self-consistent hybrid MC/RISM method is used for calculating properties of a linear polymer surrounded by colloidal particles with purely repulsive, hard-core, interactions between the particles and chain beads. Our approach combines the traditional atomistic Monte-Carlo (MC) simulation of flexible polymer chains with the numerical solution of the site-site Ornstein-Zernike-like (RISM) integral equations. Since the condensed-phase environment of a flexible macromolecule affects the equilibrium configuration probability distribution of the macromolecule, the site-site intramolecular correlation function and the intramolecular potential field are treated in a self-consistent manner. It is shown that in such an athermal system the medium-induced collapse of a polymer (similar to polymer collapse in a poor solvent) may occur. Our analysis yields a simple “entropic” interpretation of this transition. We present the detailed study of the dependence of conformational properties of the chains on the degree of polymerization, density and size of colloidal particles.

The correlation functions of hard-sphere chains: Monodisperse chains as a complete association limit

The mixture of associating hard spheres with two random association sites is considered to model freely jointed tangent hard-sphere chains of fixed length. In the case of the complete association limit with infinite association strength, the associating fluid becomes the hard-sphere chain fluid. The multidensity Ornstein–Zernike equation is applied to this limiting case, and an analytical solution is obtained within the polymer Percus–Yevick (PPY) approximation. In doing so, we imposed connectivity constraints between bonded segments in order to avoid numerically inconvenient forms. Explicit expressions for the contact values of the correlation functions are obtained, and the correlation functions for the region beyond the hard core are calculated from a set of integral equations involving only finite quantities. Predictions of the theory for 4- and 8-mer fluid are compared to computer simulation results. For overall correlation functions accurate predictions are obtained over the whole density range. For the inter- and intramolecular correlation functions, a significant improvement is found at low density compared to our previous theory with the PPY ideal-chain approximation. As chain length increases, the theory overestimates the intermolecular correlation functions, and underestimates the intramolecular correlation functions. It is concluded that the good accuracy for the overall correlation functions is due to the cancellation of errors between the inter- and intramolecular correlation functions.

Hybrid MC/RISM technique for simulation of polymer solutions: Monte Carlo+ RISM integral equations

A new (hybrid) method is reported for modelling complex macromolecular systems. The approach combines the traditional atomistic Monte Carlo (MC) computer simulation of flexible polymer chains with the numerical solution of the site-site Ornstein-Zernike-like (RISM) integral equations. The method is used for calculating properties of a linear polymer in dilute solution. Since the condensed-phase environment of a flexible macromolecule affects the equilibrium configuration probability distribution of the macromolecule, the site-site intramolecular correlation function and the intramolecular potential field are treated in a self-consistent manner. Briefly, the MC method is applied to generate the configurations of a single chain molecule. Using the coordinates of chain beads, the averaged intrapolymer correlation function is obtained. Then, solving the coupled RISM equations for a given density of solvent particles, we find the polymer-solvent correlation functions. This yields the medium-induced intrapolymer potential and the corresponding effective intramolecular energies, which are used in the standard Metropolis MC procedure. The structural properties of the polymer chain are computed by averaging over the statistically representative set of configurations. As a result of many such iterations, the intramolecular structure is determined self-consistently. Using the hybrid MC/RISM method, extensive studies have been made of static properties of flexible polymer chains surrounded by LJ particles with purely repulsive interactions between the particles and chain beads. Also, direct molecular dynamics simulations have been carried out and have demonstrated that the hybrid MC/RISM approach gives a quite accurate prediction for condensed-phase effects.

Adsorption of Polymers at the Surface Concentrations in the Diluted to Semidiluted Regimes

The polymer adsorption regimes that correspond to very diluted bulk concentration and diluted to semidiluted surface concentration (on the solid substrate) were studied both theoretically and experimentally. The extremal behavior of the adsorbed polymer layer was observed experimentally with two independent methods: (1) contact angle measurements of adsorbed polymer films of poly(nonyl acrylate) and poly(methyl methacrylate) (PMMA) on the surface of aluminum foil and (2) electrochemical reduction of K3Fe(CN)6 on the surface of a platinum electrode covered by the adsorbed polymer layer. As recently reported (Adsorp. Sci. Technol. 1996, 14, 251), the same behavior was observed on the surface of ZnO powder. The extremal behavior was found at polymer concentrations of about 0.001% in the bulk, or 1−5% from the plateau adsorption value. We suggest these data can be explained by the sharp change of conformation (and fraction of bonded units per chain) of adsorbed macromolecules when the transition from the diluted to semidiluted regime occurred. The calculations performed on the basis of the partition function combined with an associative integral equation predict the extremum in the surface coverage due to specific dependence of the intramolecular correlation function on the volume concentration of polymer.

Correlation function method for conformations of model proteins

Generalized integral equations for a complete set of intramolecular correlation functions are employed to predict the conformation of heteropolymers, e.g., proteins. The theory incorporates short- and long-range aspects of polymer interactions. Given pairwise interaction potentials between units in the chain, such as repulsive, dispersive, Coulomb and hydrogen bonding interactions, the coupled, self-consistent integral equations are shown to yield a helical form, beta form, or a folded three dimensional structure, depending on the linear sequence of the model protein molecule. Three dimensional structures for these sequences are obtained from the distance matrix via an embedding algorithm.

Quantum effects in the electronic structure of liquid methanol measured by -ray diffraction

The electronic structure factors for liquid light methanol and heavy methanol have been measured at under their normal vapour pressures using a -ray beam from an radio-isotope source. Observable differences between the electronic structure factors of the two liquids reveal small, but real changes in their r-space correlation functions. This is due to quantum effects in the intermolecular and intramolecular electron density correlation functions and will be discussed in terms of the structures of light and heavy liquid methanol.

Self-consistent integral-equation theory of chain-molecular liquids: Structure and thermodynamics

Self-consistent integral equations for the pair intramolecular and intermolecular correlation functions are derived from a general hierarchy of integral equations for chain-molecular liquids. These coupled equations are obtained by using superposition approximations for the triplet correlation functions, an approximate translational symmetry for the site–site intramolecular correlation functions and the equivalence of sites for intermolecular correlation functions. In addition to this self-consistent set of integral equations, the polymer reference interaction site model (PRISM) integral equation is also made self-consistent by coupling this intermolecular equation to the equations for the intramolecular correlation functions derived in the present theory. The intra- and intermolecular correlation functions of the self-consistent schemes considered in this work obey integral equations, and they are different from the other self-consistent schemes proposed in the literature. Self-consistent solutions for the structural properties, such as intra- and intermolecular correlation functions and structure factor, and macroscopic properties, such as chain expansion factor and thermodynamic functions of athermal polymer melts, are compared with available Monte Carlo results and other theories. For the properties examined, self-consistent solutions yield better results than the non-self-consistent calculations with ad hoc, ideal Gaussian inputs for the intramolecular correlation functions.

Thermodynamics and local structure of vinyl polymer melts

Monte Carlo simulation results are reported for the site-site pair correlations and equation of state of model vinyl polymer melts. The molecules are freely jointed hard chains with a hard sphere side-group attached to every other backbone bead. The local structure and pressure are investigated as a function of the diameter of the side group for melt-like densities. The intramolecular correlation functions are well represented by a single chain model where excluded volume interactions are included for beads separated by four bonds or less and neglected otherwise. The intermolecular correlation functions show interesting packing effects. The side group shields the backbone beads from approaching each other, to a degree that increases with increasing diameter of the side group. The polymer reference interaction site model integral equation theory is in good agreement with the simulation results for the pair correlation functions. At fixed volume fraction, the pressure is found to be a non-monotonic function of the size of the side group.

Integral equations of the correlation functions for polymeric liquids

The integral equations for intramolecular and intermolecular correlation functions are derived for nonrigid polymeric (polyatomic) liquids by the device of the Kirkwood charging parameters. These integral equations are cast into mean-field-type equations by using the potential elimination method, reported previously for dense simple fluids. Based on the mean-field integral equations, we examine the superposition approximations for various levels of correlation. The present theory provides a means to make systematic corrections for superposition approximations for correlation functions of various orders. Upon using the superposition approximations for the triplet correlation functions in the Kirkwood hierarchy and an assumption or another concerning the charging parameter dependence of the cavity functions, we derive a set of generalized Percus–Yevick and hypernetted chain integral equations for the intramolecular and intermolecular pair correlation functions for beads (sites) of polymeric (polyatomic) liquids. This set of integral equations allows the intramolecular and intermolecular correlation functions to be determined self-consistently. The connection of this set of integral equations to the bead–bead (molecular) Ornstein–Zernike relation is pointed out. The integral equations for the intramolecular correlation functions will be numerically solved for some properties of a single polymer chain in the infinite dilution limit in the sequel to this paper.

Application of the integral equation theory of polymers: Distribution function, chemical potential, and mean expansion coefficient

A recursive integral equation for the intramolecular correlation function of an isolated linear polymer of N bonds is derived from the integral equations presented in the preceding paper. The derivation basically involves limiting the density of the polymer to zero so that polymers do not interact with each other, and thus taking into account the intramolecular part only. The integral equation still has the form of a generalized Percus–Yevick integral equation. The intramolecular correlation function of a polymer of N bonds is recursively generated by means of it from those of polymers of 2, 3,..., (N−1) bonds. The end-to-end distance distribution functions are computed by using the integral equation for various chain lengths, temperatures, and bond lengths in the case of a repulsive soft-sphere potential. Numerical solutions of the recursive integral equation yield universal exponents for the mean square end-to-end distance in two and three dimensions with values which are close to the Flory results: 0.77 and 0.64 vs Flory’s values 0.75 and 0.6 for two and three dimensions, respectively. The intramolecular correlation functions computed can be fitted with displaced Gaussian forms. The N dependence of the internal chemical potential is found to saturate after some value of N depending on the ratio of the bond length to the bead radius.

Local structure of model polymeric fluids: Hard-sphere chains and the three-dimensional fluctuating bond model

Monte Carlo simulations are performed for athermal polymers in the three-dimensional fluctuating bond lattice model. Polymer molecules composed of 20 and 50 freely-jointed beads are studied at volume fractions ranging from 0.2 to 0.35. Chain dimensions, intramolecular correlation functions, and intermolecular correlation functions are compared to results of off-lattice simulations for a similar freely-jointed chain model. It is found that the intramolecular correlation functions are qualitatively similar both on and off the lattice; there are quantitative differences which arise because of the larger persistence lengths of the off-lattice chains. At low densities the intermolecular correlation functions are similar in the two models, but significant differences appear at higher densities because of the limited extent of packing in the lattice fluid. We conclude that the lattice model adequately describes the effect of configurational entropy on fluid structure, but is inadequate to address packing effects which are important in the study of polymer melts. The volumetric behavior of the two model fluids is also discussed.

A new RISM integral equation for solvated polymers

A new RISM integral equation suitable for treating solute-solvent interactions when the solute contains periodicity is presented. The essential feature is the introduction of a new intramolecular correlation function for the solute, which is a sum of the intramolecular correlation functions in the old theory over the periodic cells of the solute. The theory can be used to study polyelectrolyte properties in solution, including solvent effects on biopolymers. As a first application, simple models are considered in which atoms are arranged along a linear chain and on a helix. The results are compared with computer simulations for a linear chain of ions in water.

Numerical solution of RISM for homonuclear vibrating hard-dumbells

Numerical solutions of the reference interaction site model (RISM) for site-site radial distribution functions gαβ(r) of a fluid of vibrating harddumbells are carried out, using a method of direct substitution in Fourier space. Comparisons with molecular dynamics (MD) results for the same system are given. The intramolecular distribution functions obtained by MD are used in RISM to obtain the corresponding intermolecular site-site gαβ(r), which in turn are compared against the MD results. These numerical results and further algebraic analysis indicate that the intermolecular site-site correlation functions calculated with RISM, do not depend noticeably on the form of the intramolecular correlation function used in the calculation. Additional evidence on the equivalence of vibrating and rigid hard dumbells is given.

On the semiclassical calculation of molecular absorption and fluorescence spectra

A semiclassical expansion is developed for the direct calculation of molecular electronic (absorption and fluorescence) spectra without having to calculate the molecular vibrational eigenstates. The expansion is based on expressing the necessary dipole correlation functions (which do not have a straightforward classical analog) in terms of a hierarchy of other intramolecular correlation functions which do have a simple semiclassical expansion. To lowest order in h/ the spectra may be calculated by running classical trajectories on a zero-order Hamiltonian Hc. The present semiclassical expansion is not unique and depends on our choice of Hc. This arises since the spectra are not analytic functions of h/. Applications are made to the absorption and to the fluorescence spectra of polyatomic molecules (single- and two-photon processes). The present method may be easily generalized to other multiphoton processes.

On the derivation of rate equations for collisionless molecular multiphoton processes

Starting with the complete molecular Liouville equation and making use of the Mori–Zwanzig projection operator formalism we derive generalized rate equations which describe the collisionless dynamics of level populations in the quasicontinuum of molecular multiphoton processes. We define precisely the complete molecular information required for the description of these processes which is expressed in the form of a hierarchy of intramolecular dipole correlation functions. We then discuss the general conditions under which the equations attain the form of simple rate equations and show how the complete molecular information reduces considerably in this case to three relevant quantities per transition: an integrated Rabi frequency, a dephasing time and ratios of statistical weights of the levels.

Stochastic reduction for molecular multiphoton processes

Reduced equations of motion (REM) are derived for molecular multiphoton processes by constructing a small set of relevant operators corresponding to populations and coherences of the molecule ’’dressed’’ by the radiation field and adopting the Mori projection operator technique. The resulting REM contain time-dependent dephasing operators which are intimately related to our reduction scheme and whose definition and evaluation do not rely on any perturbative expansion in intramolecular interactions. These operators are expressed in terms of intramolecular dipole correlation functions which are the key quantities in the present formulation and are then evaluated using a simple model. In the limit where our physical system may be separated into a ’’system’’ and a ’’bath’’ with a weak interaction, we recover the ordinary expressions of line broadening. The present formulation provides us with a hiearchy of REM which interpolate smoothly all the way from the coherent to the incoherent driving limits and are in good quantitative agreement with the existing experimental data.

Reduced Equations of Motion for Molecular Multiphoton Processes

I derive reduced equations of motion for molecular multiphoton processes by constructing a set of relevant operators and adopting the Mori projection-operator technique. The intramolecular dephasing processes which play a dominant role in the quasicontinuum are properly incorporated by utilizing the representation of exact molecular states. The resulting equations include dephasing operators expressed in terms of intramolecular dipole correlation functions, and interpolate between the coherent and incoherent driving limits.