Carol K. Hall

High density Monte Carlo simulations of chain molecules: Bulk equation of state and density profile near walls

We introduce a new Monte Carlo method suitable for simulations of chain molecules over a wide range of densities. Results for the equation of state of chains composed of 4, 8, and 16 freely joined hard spheres are compared with the predictions of several theories. The density profile of the fluid in the vicinity of the wall, and the scaling of the pressure with chain length are also discussed.

A new equation of state for athermal chains

A new equation of state for fluids containing athermal chain molecules is developed and compared to simulation results and existing theories in three, two, and one dimensions. The new expression, which builds upon the generalized Flory theory, contains no adjustable parameters and relates the compressibility factor of an n‐mer fluid to the compressibility factors of monomer and dimer fluids at the same volume fraction. Comparisons with Monte Carlo results for three‐ and two‐dimensional freely jointed chains show very good agreement, and the overall accuracy of the new expression appears comparable to Wertheim’s thermodynamic perturbation theory of polymerization. In one dimension the new expression reduces to the exact result. Application of the equation to chain models with internal constraints and overlapping hard sites is discussed and illustrated through comparisons with Monte Carlo results for rigid trimers. An extension of our approach to arbitrary reference fluids shows that the generalized Flory a...

Equation of state for chain molecules: Continuous‐space analog of Flory theory

New, accurate equations of state for fluids of chain molecules are derived as generalizations of the well‐known Flory and Flory–Huggins lattice theories to continuous space. Comparison with the results of new Monte Carlo simulations of athermal chains (freely jointed hard disks and spheres), extending over a wide range of densities, reveals that the generalized Flory–Huggins equation of state provides an accurate prediction for the pressure.

Equilibrium thermodynamics of homopolymers and clusters: Molecular dynamics and Monte Carlo simulations of systems with square-well interactions

The thermodynamics of homopolymers and clusters with square-well interactions of up to 64 particles are studied with constant-temperature discontinuous molecular dynamics ~DMD! simulations; for comparison Monte Carlo ~MC! simulations are also reported. Homopolymers composed of more than five beads are found to exhibit two or more equilibrium transitions. In the long chain limit, these multiple transitions correspond to gas-to-liquid, liquid-to-solid, and solid-to-solid transitions. In particular, the liquid-to-solid-like disorder-to-order transition for isolated 32mers and 64mers is strongly first order ~bimodal energy distribution! at the reduced square-well diameter l51.5. As l decreases from 1.5 to 1.3, the bimodal distribution becomes unimodal. The use of Lindemann’s rule for solids indicates that the structure formed right below the liquid-to-solid transition temperature has a solid core but a liquid surface. Comparing the homopolymer results with those for square-well clusters indicates that the bonding constraint in homopolymers increases the temperatures of transitions but decreases their strength. The solid structure of an isolated 64mer is nearly identical to that of a cluster of 64 beads. Possible approaches to the experimental observation of the solid-state for an isolated chain are discussed.

α‐Helix formation: Discontinuous molecular dynamics on an intermediate‐resolution protein model

An intermediate‐resolution model of small, homogeneous peptides is introduced, and discontinuous molecular dynamics simulation is applied to study secondary structure formation. Physically, each model residue consists of a detailed three‐bead backbone and a simplified single‐bead side‐chain. Excluded volume and hydrogen bond interactions are constructed with discontinuous (i.e., hard‐sphere and square‐well) potentials. Simulation results show that the backbone motion of the model is limited to realistic regions of Φ–Ψ conformational space. Model polyalanine chains undergo a locally cooperative transition to form α‐helices that are stabilized by backbone hydrogen bonding, while model polyglycine chains tend to adopt nonhelical structures. When side‐chain size is increased beyond a critical diameter, steric interactions prevent formation of long α‐helices. These trends in helicity as a function of residue type have been well documented by experimental, theoretical, and simulation studies and demonstrate the ability of the intermediate‐resolution model developed in this work to accurately mimic realistic peptide behavior. The efficient algorithm used permits observation of the complete helix–coil transition within 15 min on a single‐processor workstation, suggesting that simulations of very long times are possible with this model. Proteins 2001;44:344–360.

Site–site correlations in short chain fluids

Intramolecular and intermolecular site–site correlations in short chain fluids are obtained via Monte Carlo simulation for volume fractions ranging between 0.05 and 0.35. The chains are modeled as pearl necklaces of freely jointed hard spheres; chains composed of 4 and 8 beads are studied. The intramolecular distribution between a pair of beads separated by a fixed number of segments along the chain is found to be remarkably independent of the position of the pair along the chain. At low densities the intermolecular site–site pair distribution function at contact is found to be much less than one due to the ‘‘correlation hole’’ effect. The contact value increases as the density is increased, and decreases as the chain length is increased. We use the intramolecular correlations measured to obtain polymer reference interaction site model predictions for the intermolecular site–site distribution function. We find that the theory accurately reproduces the local structure of the fluid, but significantly overestimates the contact value of the distribution function, especially at low densities. A comparison of freely jointed chain results with simulations of chains with fixed bond angles and torsional rotations treated in the rotational isomeric state approximation shows that the correlation hole is more pronounced in freely jointed chains. We test a superposition approximation used to evaluate the three body term in the pressure equation for chain molecules. We find that the three‐body term is sizeable, and that the superposition approximation significantly underestimates the three‐body contribution.

Influence of polymer molecular weight and temperature on phase composition in aqueous two-phase systems

Abstract In an effort to understand the molecular weight and temperature dependence of protein partition coefficients, we have conducted experiments to determine how the binodal curve and the difference in polymer concentrations between the top and bottom phases (▵ PEG  [PEG] T — [PEG] B and Δ Dx  [Dx] B — [Dx] T ); depend on polymer molecular weight and temperature. We report the equilibrium compositions at 4°, 25° and 40°C for 16 different aqueous two-phase PEG-Dx systems which consist of all possible combinations of the four PEG molecular weights (4000, 6000, 10 000, and 20 000) and the four Dx molecular weights (10 000, 40 000, 110 000 and 500 000). Four tie lines were measured for each system. We confirm that increasing polymer molecular weight increases the tie line length and depresses the binodals to lower polymer concentrations, and that this effect will be greater when the ratio of this molecular weight increase to the molecular weight of the other species is large. The increase in the tie line length and in Δ Dx and Δ PEG with increasing polymer molecular weight tends to level off at high polymer molecular weight and at high total polymer concentrations. It is shown that the effect of temperature is not symmetric, i.e. changes in temperature produce a change in Δ Dx while Δ PEG remains nearly constant. The effect of polymer molecular weight on both Δ Dx and Δ PEG is magnified with increasing temperature. The results found here suggest a series of experiments which could be used to isolate (or highlight) the effect of the second virial coefficient on the partition coefficient.

Monte Carlo simulations and integral equation theory for microscopic correlations in polymeric fluids

Monte Carlo simulations are performed for polymer chains modeled as pearl necklaces of freely jointed tangent hard spheres; chains composed of 20, 50, and 100 beads are studied at volume fractions ranging from 0.1 to 0.35. The mean‐square end‐to‐end distance, the radius of gyration, and the intramolecular and intermolecular site–site distribution functions are monitored in the simulations. Various approximations for the intramolecular structure factor, w(k), are tested. It is found that the w(k) from the semi‐flexible chain model is the most accurate. The polymer ‘‘reference interaction site model’’ (PRISM) theory of Curro and Schweizer is tested using both approximate and exact expressions for w(k). It is found that, at the densities examined here, the theory is accurate for the local structure except near contact where it tends to overestimate the value of the intermolecular site–site distribution function, g(r). The polymer‐RISM theory is also solved with the generalized mean spherical approximation...

Monte Carlo simulation of hard chain–hard sphere mixtures in slitlike pores

The structure of mixtures of hard chains (modeled as a pearl necklace of freely jointed hard spheres) and hard spheres in slitlike pores is studied using a canonical ensemble Monte Carlo method. Simulation results for the density profiles in pores with wall separations varying from two to ten hard sphere diameters are presented at overall volume fractions of 0.12 and 0.34, and the effect of pore size, chain concentration, and chain length on the structure of the chains is investigated. It is found that the chains are depleted at the wall at the lower density, but enhanced at the wall (relative to the center of the pore) at the higher density; this depletion increases as the chain length is increased. It is found that the enhancement of monomers and depletion/enhancement of chains at the wall becomes more marked as the pore size is increased. As all the pores we study are integer multiples of the bead diameter, we do not observe the oscillatory variation of the wall density with wall separation which is ex...

Generalized Flory equations of state for square‐well chains

The hard chain Dickman–Hall generalized Flory (GF) and Honnell–Hall generalized Flory‐dimer (GF‐D) equations of state are extended to square‐well chain fluids. The molecules are modeled as a pearl necklace of freely jointed spheres that interact via site–site square‐well intermolecular potentials. Equations of state for square‐well monomers and square‐well dimers (required in the GF‐D theory) are obtained from integral equations with a mean spherical approximation (MSA) closure. The theories are compared to Monte Carlo simulation data for the pressure of square‐well 4‐mers, 8‐mers, and 16‐mers. The GF‐D theory is in excellent agreement with the simulation data; the GF theory overestimates the pressure in all cases. A closed‐form equation of state for square‐well chains is obtained by employing equations of state for square‐well monomers and for square‐well dimers using second order perturbation theory. The resulting equation is very accurate when compared to simulations, but not as accurate as when the mo...