A tiered approach to Monte Carlo sampling with self-consistent field potentials

A "tiered" approach to Monte Carlo sampling of nuclear configurations is presented for ab initio, self-consistent field (SCF)-based potentials, including Hartree-Fock and density functional theory. Rather than Metropolis testing only the final SCF energy, individual cycle energies are tested in a tiered fashion, without approximation. Accordingly, rejected configurations are terminated early in the SCF procedure. The method is shown to properly obey detailed balance, and effective modifications are presented for cases in which the initial SCF guess is particularly poor. Demonstrations on simple systems are provided, including an assessment of the thermal properties of the neutral water dimer with B3LYP/6-31++G**. Cost analysis indicates a factor-of-two reduction in SCF cycles, which makes the method competitive with accelerated molecular dynamics sampling techniques, without the need for forces.     

Star-branched polymers in an adsorbing slit: a Monte Carlo study

A coarse-grained model of star-branched polymer chains confined in a slit was studied. The slit was formed by two parallel impenetrable surfaces, which were attractive for polymer beads. The polymer chains were flexible homopolymers built of identical united atoms whose positions in space were restricted to the vertices of a simple cubic lattice. The chains were regular star polymers consisted of f = 3 branches of equal length. The chains were modeled in good solvent conditions and, thus, there were no long-range specific interactions between the polymer beads-only the excluded volume was present. Monte Carlo simulations were carried out using the algorithm based on a chain's local changes of conformation. The influence of the chain length, the distances between the confining surfaces, and the strength of the adsorption on the properties of the star-branched polymers was studied. It was shown that the universal behavior found previously for the dimension of chains was not valid for some dynamic properties. The strongly adsorbed chains can change their position so that they swap between both surfaces with frequency depending on the size of the slit and on the temperature only.     

Using Monte Carlo simulations and self-consistent field theory to design interfacially active copolymers

We use both Monte Carlo computer simulations and numerical self-consistent field lattice calculations to determine the behavior of copolymers at penetrable and impenetrable interfaces. These computational techniques are useful as “design tools”: they allow us to systemically vary the copolymer architecture, determine optimal structures for specific applications, and establish guidelines for fabricating copolymers that yield the desired interfacial properties. We illustrate this principle with three different examples. In the first study, we combine the techniques to design copolymer compatibilizers that enhance the strength of immiscible polymer blends. These copolymers contain teeth that associate across the penetrable interface between the phase-separated regions and form a “molecular velcro” that effectively binds the regions together. In the case of impenetrable interfaces, we determine how the copolymer sequence distribution affects the structure of a layer of copolymers grafted onto a solid surface. The results indicate how to control the morphology of the layer and the surface properties of the substrate, by varying the microstructure of the grafted copolymers. Finally, we design a polymer channel that “opens” and “closes” in response to changes in the pH and quality of the surrounding solvent. The channel is formed from polyacid chains that are anchored onto a solid surface. Due to these properties, the system can be used for controlled release or sensor devices.     

Statistics of stiff polymer chains near an adsorbing surface

The exact solution of the problem of adsorption of a long ideal polymer chain with variable degree of stiffness on a plane surface is presented. It is shown that the adsorption of stiff polymer chains is a second-order phase transition; in the adsorbed state “train” (i.e. adsorbed) sections are relatively longer and loop sections relatively shorter than for flexible chains. This effect is very pronounced: already for moderately stiff chains the number of Kuhn segment lengths in one “train” section at the temperature T = T_cr/2 (T_cr is the critical temperature for adsorption transition) can reach several thousands, and deviation from the surface occurs only in the form of small “hairpins”. The maximum length of the chain, which at the given conditions would flatten completely on the surface, is estimated.     

The model of a polyelectrolyte solution with explicit account of counterions within a self-consistent field theory

A theoretical approach to calculating the thermodynamic and structural functions of solutions of polyelectrolytes based on Gaussian equivalent representation for the calculation of functional integrals is proposed. It is noted that a new analytical result of this work is the direct assumption of counterions, along with an equation for the gyration radius of a polymer chain as a function of the concentrations of monomers and added low-molecular salt. An equation of state is obtained within the proposed model. Our theoretical results are used to describe the thermodynamic and structural properties of an aqueous solution of sodium polystyrene sulfonate with additions of NaCl.     

Development of the theory of a self-consistent field for polyelectrolyte solutions

A new theoretical approach to calculating the thermodynamic and structure functions of polyelectrolyte solutions is proposed, based on the method of Gaussian equivalent representation for calculating the functional integrals. Formulas for the mean-force potential, osmotic pressure, and complete monomer-monomer pair distribution functions are presented. A sodium polystyrene sulfonate solution with NaCl additives is considered as an example.     

Monte Carlo model of the temperature rise at a GaAs surface under an electron beam

Scanning electron microscopy (SEM) has frequently been used to study semiconductor materials. It offers the possibility of obtaining reliable qualitative and quantitative information on relevant local material parameters. The temperature rise due to electron‐beam bombardment can influence some semiconductor parameters, which then will influence the SEM information. In this work we propose a model calculation based on the Monte Carlo (MC) method to calculate the temperature rise due to electron‐beam heating. The results show that the temperature rise increases with increasing numbers of electrons (electron‐beam current), and the inverse behavior is observed with respect to the electron energy (electron‐beam voltage). The decrease in temperature rise with depth is also obtained. Copyright © 2006 John Wiley & Sons, Ltd.     

Static and dynamic properties of tethered chains at adsorbing surfaces: a Monte Carlo study

We present extensive Monte Carlo simulations of tethered chains of length N on adsorbing surfaces, considering the dilute case in good solvents, and analyze our results using scaling arguments. We focus on the mean number M of chain contacts with the adsorbing wall, on the chain's extension (the radius of gyration) perpendicular and parallel to the adsorbing surface, on the probability distribution of the free end and on the density profile for all monomers. At the critical adsorption strength epsilon(c) one has M(c) approximately N(phi), and we find (using the above results) as best candidate phi to equal 0.59. However, slight changes in the estimation of epsilon(c) lead to large deviations in the resulting phi; this might be a possible reason for the difference in the phi values reported in the literature. We also investigate the dynamical scaling behavior at epsilon(c), by focusing on the end-to-end correlation function and on the correlation function of monomers adsorbed at the wall. We find that at epsilon(c) the dynamic scaling exponent a (which describes the relaxation time of the chain as a function of N) is the same as that of free chains. Furthermore, we find that for tethered chains the modes perpendicular to the surface relax quicker than those parallel to it, which may be seen as a splitting in the relaxation spectrum.     

Monte Carlo approach to the decay rate of a metastable system with an arbitrarily shaped barrier

A path integral Monte Carlo method based on the fast-Fourier transform technique combined with the important sampling method is proposed to calculate the decay rate of a metastable quantum system with an arbitrary shape of a potential barrier. The contribution of all fluctuation actions is included which can be used to check the accuracy of the usual steepest-descent approximation, namely, the perturbation expansion of potential. The analytical approximation is found to produce the decay rate of a particle in a cubic potential being about 20% larger than the Monte Carlo data at the crossover temperature. This disagreement increases with increasing complexity of the potential shape. We also demonstrate via Langevin simulation that the postsaddle potential influences strongly upon the classical escape rate.     

Self-assembly of ABC star triblock copolymer thin films confined with a preferential surface: a self-consistent mean field theory

The microphase separation and morphology of a nearly symmetric A(0.3)B(0.3)C(0.4) star triblock copolymer thin film confined between two parallel, homogeneous hard walls have been investigated by self-consistent mean field theory (SCMFT) with a pseudospectral method. Our simulation experiments reveal that under surface confinement, in addition to the typically parallel, perpendicular, and tilted cylinders, other phases such as lamellae, perforated lamellae, and complex hybrid phases have been found to be stable, which is attributed to block-substrate interactions, especially for those hybrid phases in which A and B blocks disperse as spheres and alternately arrange as cubic CsCl structures, with a network preferred structure of C block. The results show that these hybrid phases are also stable within a broad hybrid region (H region) under a suitable film thickness and a broad field strength of substrates because their free energies are too similar to being distinguished. Phase diagrams have been evaluated by purposefully and systematically varying the film thickness and field strength for three different cases of Flory-Huggins interaction parameters between species in the star polymer. We also compare the phase diagrams for weak and strong preferential substrates, each with a couple of opposite quality, and discuss the influence of confinement, substrate preference, and the nature of the star polymer on the stability of relatively thinner and thick film phases in this work.     

Monte Carlo simulation of an inhomogeneous dielectric continuum model for B-DNA

Thermodynamic and structural properties of the counterion atmosphere surrounding B-DNA are calculated by Monte Carlo simulation in a spatially inhomogeneous, but piecewise uniform, dielectric continuum cell model - the "barbarous" model. A boundary element formulation is implemented to study the sensitivity of these properties with respect to perturbations in the location of discontinuous dielectric boundaries relative to fixed and mobile charges. High concentrations are considered corresponding to the liquid crystalline hexagonally ordered phase of DNA. Primitive model results are verified against other simulation reports and a comparison of barbarous model predictions with experimental data is discussed. The internal energy, osmotic coefficient, radial distributions and the population ratio of counterions in the geometrically resolved major and minor grooves are all found to strongly depend on the dielectric boundary position. This suggests that a self-consistent development of the model should consider a free surface problem where the boundary is not specified a priori.     

Counterion localization in solutions of starlike polyelectrolytes and colloidal polyelectrolyte brushes: a self-consistent field theory

A quantitative analysis of the distribution of counterions in salt-free solutions of colloidal polyelectrolyte brushes and starlike polyelectrolytes is performed on the level of the Poisson-Boltzmann approximation. Exact numerical solutions are obtained for starlike polyelectrolyte molecules composed of f = 20, . . ., 50 arms with a fixed fractional charge alpha per segment by applying the self-consistent field method of Scheutjens and Fleer (SF-SCF). The Wigner-Seitz cell dimension defines the concentration of polyelectrolyte stars in the system. The numerical results are compared to predictions of an analytical mean field theory and related to experimental observations on the osmotic pressure in solutions of starlike polyelectrolytes and colloidal polyelectrolyte brushes.     

Adsorption of random copolymers by a selective layer: Monte Carlo studies

We use scaling arguments and computer simulations to investigate the adsorption of symmetric AB-random copolymers (RC) from a diluted solution onto a selective ABA layer. Depending on the ratio between the layer thickness and the size of excess blobs, d/xi, three regimes of RC adsorption are predicted. For large values of the layer thickness RC adsorption can be understood as adsorption on two selective interfaces where sequences of RC chains form bridges. When the layer thickness is of the order of xi, excess blobs are trapped in the layer and localize the copolymer chain strongly. If the layer thickness is very small a weak adsorption scenario is predicted where large loops are formed outside the layer. Our simulations using the bond fluctuation model are in good agreement with the scaling predictions. We show that chain properties display non-monotonous behavior with respect to the layer thickness with optimal values for d approximately xi. In particular, we discuss simulation results for density profiles, statistics of bridges, loops and tails formed by the adsorbed chains, as well as for the adsorption order parameter and free energy.     

Adsorption of molecular brushes with polyelectrolyte backbones onto oppositely charged surfaces: a self-consistent field theory

The two-gradient version of the Scheutjens-Fleer self-consistent field (SF-SCF) theory is employed to model the interaction between a molecular bottle brush with a polyelectrolyte backbone and neutral hydrophilic side chains and an oppositely charged surface. Our system mimics graft-copolymers with a cationic main chain (among which poly( L-lysine)- graft-poly(ethylene glycol) (PLL- g-PEG) or poly( L-lysine)- graft-polyoxazoline are well-known examples) interacting with negatively charged (metal oxide) solid surfaces. We aim to analyze the copolymer-surface interaction patterns as a function of the molecular architecture parameters. Two regimes are investigated: First, we compute the effective interaction potential versus the distance from the surface for individual bottle brush macromolecules. Here, depending on the grafting ratio and the degree of polymerization of the side chains, the interplay of electrostatic attractions of the main chain to the surface and the steric repulsion of the grafts results in different patterns in the interaction potential and, therefore, in qualitatively different adsorption behavior. In particular, we demonstrate that, at high side chain grafting density and short grafts, the molecular brushes are strongly adsorbed electrostatically onto negatively charged substrates, whereas, in the opposite case of low grafting ratio and high molecular weight of grafts, the steric repulsion of the side chains from the surface dominates the polymer-surface interaction. At intermediate grafting ratios, the adsorption/depletion scenario depends essentially on the ratio between the electrostatic screening length and the thickness of the molecular bottle brush. We further have analyzed the equilibrium adsorbed amount as a function of the macromolecular architecture. Our results are based on a detailed account of attractive and repulsive (including intermolecular in-plane) interactions, and suggest a nonmonotonic dependence of the adsorbed amount on the grafting ratio, in good agreement with the experimental studies for PLL- g-PEG adsorption onto Nb2O5 surfaces. The results of the theoretical modeling are discussed in the context of the optimization of the PLL-g-PEG molecular design parameters in order to create protein-resistant surfaces.     

Monte Carlo simulations of a coarse grained model for an athermal all-polystyrene nanocomposite system

The structure of a polystyrene matrix filled with tightly cross-linked polystyrene nanoparticles, forming an athermal nanocomposite system, is investigated by means of a Monte Carlo sampling formalism. The polymer chains are represented as random walks and the system is described through a coarse grained Hamiltonian. This approach is related to self-consistent-field theory but does not invoke a saddle point approximation and is suitable for treating large three-dimensional systems. The local structure of the polymer matrix in the vicinity of the nanoparticles is found to be different in many ways from that of the corresponding bulk, both at the segment and the chain level. The local polymer density profile near to the particle displays a maximum and the bonds develop considerable orientation parallel to the nanoparticle surface. The depletion layer thickness is also analyzed. The chains orient with their longest dimension parallel to the surface of the particles. Their intrinsic shape, as characterized by spans and principal moments of inertia, is found to be a strong function of position relative to the interface. The dispersion of many nanoparticles in the polymeric matrix leads to extension of the chains when their size is similar to the radius of the dispersed particles.     

Hybrid Monte Carlo — Fluid model for studying the effects of nitrogen addition to argon glow discharges

A computer model is developed for describing argon/nitrogen glow discharges. The species taken into account in the model include electrons, Ar atoms in the ground state and in the 4s metastable levels, N₂ molecules in the ground state and in six different electronically excited levels, N atoms, Ar⁺ ions, N⁺, N₂⁺, N₃⁺ and N₄⁺ ions. The fast electrons are simulated with a Monte Carlo model, whereas all other species are treated in a fluid model. 74 different chemical reactions are considered in the model. The calculation results include the densities of all the different plasma species, as well as information on their production and loss processes. The effect of different N₂ additions, in the range between 0.1 and 10%, is investigated.     

On the mechanism of uptake of globular proteins by polyelectrolyte brushes: a two-gradient self-consistent field analysis

We present model calculations for the interaction of a protein-like inhomogeneously charged nanoscale object with a layer of densely grafted polyelectrolytes ("polyelectrolyte brush"). The motivation of this work is the recent experimental observation that proteins that carry an overall negative charge are absorbed into negatively charged polyelectrolyte brushes. Two-gradient self-consistent field (2G-SCF) calculations have been performed to unravel the physical mechanism of the uptake of protein thus effected. Our results prove that an overall neutral, protein-like object can electrostatically be attracted and therefore spontaneously driven into a polyelectrolyte brush when the object has two faces (patches, domains), one with a permanent positive charge and the other with a permanent negative charge. Using a 2G-SCF analysis, we evaluate the free energy of insertion, such that the electric dipole of the inclusion is oriented parallel to the brush surface. An electroneutral protein-like object is attracted into the brush because the polyelectrolyte brush interacts asymmetrically with the charged patches of opposite sign. At high ionic strength and low charge density on the patches, the attraction cannot compete with the repulsive excluded-volume interaction. However, for low ionic strengths and sufficiently high charge density on the patches, a gain on the order of k(B)T per charge becomes possible. Hence, the asymmetry of interaction for patches of different charges may result in a total attractive force between the protein and the brush. All results obtained herein are in excellent agreement with recent experimental data.     

Coupling quantum Monte Carlo to a nonlinear polarizable continuum model for spherical solutes

Starting from the nonlinear dielectric response model of Sandberg and Edholm, we derive an analytical expression of the polarization contribution to the solvation free energy in terms of the electronic density of the solute and the dielectric properties of the solvent. The solvent inhomogeneity is taken into account with the use of a smooth switching function whose spacial variation is established on the basis of how the solvent is arranged around the solute. An explicit form of a local potential representing the solvent effect on the solute is thus obtained by functional analysis. This effective potential can be combined with density functional or quantum chemical methods for the quantum mechanical treatment of the solute. Here, we use quantum Monte Carlo techniques for the solute and apply the method to the hydration of atomic ions finding very good agreement with experimental data.     

Monte Carlo simulations of a polyelectrolyte chain with added salt: effect of temperature and salt valence

Using the cooperative motion algorithm, the effect of salt valence z(s) and of the reduced temperature T* on a single polyelectrolyte chain as well as on counterions and salt ions themselves is studied. The calculations show that both parameters strongly influence the polymer, causing it to undergo conformational changes. For a given number of the added salt cations (anions) n(s) and temperature T*, the chain takes more and more compact forms as z(s) increases (z(s) > 0). For fixed z(s), in turn, the polymer size reduces sharply as T* drops down from intermediate to low. For high T* configurational the entropy dominates the chain statistics and the mean-square radius of gyration (s2)1/2(T*,n(s),z(s)) approaches its athermal value. The low-temperature polymer collapse is also accompanied by a drop in the effective mean charge per monomer q*(T*,n(s),z(s)) (condensation of ions onto the chain) and the total inner energy e*(T*,n(s),z(s)). Furthermore, the local structure of the system is analyzed by means of pair-correlation functions g(ab)(r,T*,n(s),z(s)). At lower T* they possess sharp local maxima at small interparticle distances r that disappear as T* grows. The former observation indicates that at lower T* the ions tend to group themselves close to each other. In particular, it is concluded that the condensation is dominated by the multivalent salt ions carrying charges of opposite sign to that of monomers.     

Monte Carlo simulation of low–medium energy electrons backscattered from C/Al double layer thin films

The backscattering coefficient of low–medium energy electron beams (from 250 to 10 000 eV) impinging on C/Al double layered thin films was investigated by a Monte Carlo simulation. The aim of the research was to study the behaviour of the backscattering coefficient as a function of the beam primary energy and the thicknesses of the two layers. The backscattering coefficient as a function of the primary energy presents features that can be used to evaluate the thicknesses of the two layers. Copyright © 2006 John Wiley & Sons, Ltd.