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Phase transitions of simple fluids and binary fluid mixtures confined into long cylindrical pores are re-examined, such as capillary condensation/evaporation and wetting transitions. While a large part of the literature ignores the fact that due to the quasi-one-dimensional character of these systems a singular behavior associated with a sharp phase transition cannot occur, we pay attention to the extent in which these phase transitions are smoothed out (in relation to the magnitude of the pore cross-sectional area). We argue that the finiteness of the pore length is an important parameter which controls the physical phenomena that are observed in simulations (and presumably also experiments explaining the distinction between the apparent “pore critical temperature” and the “hysteresis critical temperature”). We illustrate our arguments with recent findings from simulations of a lattice gas/Ising system and of the Asakura-Oosawa model of colloid-polymer mixtures.

For an understanding of interfacial phenomena of fluids on the nanoscale a detailed knowledge of the excess free energies of fluids due to walls is required, as well as of the interfacial tension between coexisting fluid phases. A description of simulation approaches to solve this task is given for a suitable model binary (A + B) fluid. Sampling the order parameter distribution of the system without walls, the curvature dependent and flat interfacial tensions of coexisting ‘bulk’ phases is extracted. In a thin film geometry, the difference in wall free energies is found via a new thermodynamic integration method. Thus the contact angle θ of macroscopic droplets is estimated from Young's equation, for varying interactions between the fluid particles and the walls, which compares well with direct observations of inclined interfaces in ultrathin slit pores. Studying two-phase situations where a wall-attached droplet exists in the slit pore in thermal equilibrium, the excess free energy due to the droplet is ...

Bottle-brush polymers contain a long flexible macromolecule as a backbone to which flexible side chains are grafted. Through the choice of the grafting density and the length of the side chains the local stiffness of this cylindrical molecular brush can be controlled, but a quantitative understanding of these phenomena is lacking. Monte Carlo simulation results are presented and discussed which address this issue, extractingmesoscopic length scales (such as the cross-sectional radius, persistence length, and contour length of these objects). Large-scale simulations of the bond fluctuation model are combined with simulations of the simple selfavoiding walk (SAW) model with flexibility controlled by a bond-angle potential, using the pruned-enriched Rosenbluth algorithm. It is shown that under good solvent conditions the bottle-brush polymers never display a pre-asymptotic Gaussian regime that would be described by the Kratky–Porod worm-like chain model, unlike the semiflexible SAWmodel. Implications of these results for the proper interpretation of experiments are discussed.

We consider the ejection dynamics of a flexible polymer chain out of confined environment. This situation arises in different physical contexts, including a flexible synthetic polymer partially confined in a nanopore and a viral genome partially ejected from its capsid. We describe the chain release from confinement both analytically and by means of dynamic Monte Carlo simulation. We find two distinct regimes of ejection dynamics depending on whether the chain is fully or partially confined. Partially confined chains are ejected from a pore of length L and diameter D after a typical time τ ∝ L2D5/3, regardless of their contour length N. The process is driven by a constant force f ≈ 5kBT/D and follows a “capillary” law. The force value is model-independent as long as the pore diameter exceeds the persistence length of the polymer chain and for pore walls that do not attract the segments of the polymer. In contrast, the ejection of fully confined chains is largely diffusive, the residence time being a nonmo...

A coarse grained model of a flexible macromolecule end-grafted on the inside of a sphere or a cylinder under good solvent conditions is studied by Monte Carlo simulations. For cylindrical confinement, two regimes are found: when the cylinder radius R exceeds the gyration radius R 90 of the polymer mushroom grafted to a planar surface, a simple scaling description holds. In the opposite case, a non-monotonic crossover to a cigar-like quasi-one-dimensional structure occurs, and the distribution P e (x) of the free chain end in the x-direction along the cylinder axis becomes bimodal. Spherical confinement, on the other hand, causes a crossover from dilute to semidilute behavior of the structural properties.

Conformations of a bottle-brush polymer with two types (A,B) of grafted side chains are studied by molecular dynamics simulations, using a coarse-grained bead−spring model with side chains of up to...

Extensive Monte Carlo simulations are presented for bottle-brush polymers under good solvent conditions, using the bond fluctuation model on the simple cubic lattice. Varying the backbone length (from Nb = 67 to Nb = 259 effective monomers) as well as the side chain length (from N = 6 to N = 48), for a physically reasonable grafting density of one chain per backbone monomer, we find that the structure factor describing the total scattering from the bottle-brush provides an almost perfect match for some combinations of (Nb, N) to experimental data of Rathgeber et al. [J. Chem. Phys. 2005, 122, 124904], when we adjust the length scale of the simulation to reproduce the experimental gyration radius of the bottle-brush. While in the experiment other length scales (gyration radius of side chains, backbone persistence length, scale characterizing the radial monomer density profile in the plane normal to the backbone) can be extracted only via fitting to a complicated and approximate theoretical expression deriv...

The finite size scaling analysis of Monte Carlo data is discussed for two models for which hyperscaling is violated: (i) the random field Ising model (using a model for a colloid-polymer mixture in a random matrix as a representative) (ii) The Ising bi-pyramid in computing surface fields.

The spontaneous rise of a fluid in a brush-coated nanocapillary is studied by molecular dynamics simulation of a coarse-grained model. The cases of changing wettability of both the capillary walls and the brush were examined. We also investigated the impact of polymer chain length on the transport of fluid along the nanotube. We found that capillary filling takes place in both lyophilic and lyophobic tubes, provided that the polymer brush coating is wetted by the fluid. In all the cases studied, capillary rise proceeds by a time-square law, but the mechanisms behind them (Lucas-Washburn or diffusive propagation) differ, depending on the chain length N. For a wettable wall, the speed of fluid imbibition decreases steadily with growing N, whereas the meniscus speed goes through a minimum at intermediate chain lengths. The polymer brush coating reorganizes into "channels" parallel to the tube axis and forms a dense plug of monomers in the vicinity of the meniscus, which moves with the meniscus along the nanotube. For lyophobic capillary walls (covered with a wettable polymer brush), depending on the chain length N, one finds three regimes: (1) short chains--one observes no meniscus motion, but an influx of fluid through the wet brush; (2) intermediate chain lengths--the fluid creates "fluid walls" inside the brush by diffusive spreading, whereby a meniscus is formed and moves up within the fluid walls; and (3) long chains--a "negative curvature" meniscus rises up the capillary by means of diffusive propagation.

The phase diagram of polymer solutions (cf. e.g. alkanes dissolved in supercritical carbon dioxide) is complicated, since there are four control parameters (temperature, pressure, monomer volume fraction, chain length of the polymer) and due to the interplay of liquid-vapor transitions and fluid-fluid unmixing. As a result I very intricate phase diagram topologies can result. An attempt to develop coarse-1 grained models that can deal with this task will be described. As usual, the polymers I will be modelled as off-lattice bead-spring chains, where several chemical monomers I are integrated into one effective bond, torsional degrees of freedom being dis-I regarded. But also a coarse-grained description of the solvent is needed: we show that using a simple point particle with a quadrupole moment and suitable Lennard-Jones I interaction yields a very good description of pure carbon dioxide (better than fully atomistic models with potentials from ab initio quantum chemistry calculations). The strength of the quadrupole moment is taken from experiment, and the Lennard-Jones parameters are adjusted such that the experimental critical temperature and density is correctly reproduced. This procedure works for other solvents as well, such as benzene. The pure alkane phase diagram is also reproduced by similarly chosen Lennard-Jones (LJ) potentials among the monomers, and the monomer-carbon dioxide LJ parameters are chosen from Lorentz-Berthelot mixing rules. These parameters are then used as input both for Monte Carlo calculations and approximate methods to calculate the equation of state, such as density functional and perturbation theory methods. Apart from the region close to critical points, where mean-field type methods fail, the analytical calculations agree well with the simulations. The solvent model, in which quadrupolar interactions are explicitly considered, is important in order to obtain a fair agreement with mixture experimental results.

Thin Ising films with nearest-neighbor ferromagnetic exchange in a bulk magnetic field H are studied in a L×L×D geometry, where at the opposite walls, given by the L×L surfaces, local magnetic fields H 1, and H D act. While in previous work, the symmetric case H 1=H D (leading to “capillary condensation”, when one applies the lattice gas terminology) as well as the antisymmetric case H 1=−H D (leading to “interface localization transitions”) were studied, we focus here on the general ‘asymmetric’ case. Monte Carlo simulations are carried out and analyses based on thermodynamic integration methods are used to establish the phase diagrams and study the properties of the coexisting phases. A discussion is given why for the range of thicknesses that is explored (16≤D≤80 lattice spacings) this is the most suitable methodology. Restricting attention to cases where in the semi-infinite system a first-order wetting transition occurs, it is shown that the latter, due to confinement, is turned in a thin-film triple point. Above the triple point, narrow two-phase coexistence curves are found, which are the analog of prewetting transitions in the semi-infinite system. A comparison to related results for (symmetrical) polymer blends and (asymmetric) colloid-polymer mixtures is made.

A recently introduced coarse-grained pair potential for carbon dioxide molecules is used to compute structural properties in the supercritical region near the critical point, applying Monte Carlo simulations. In this model, molecules are described as point particles, interacting with Lennard-Jones (LJ) forces and a (isotropically averaged) quadrupole–quadrupole potential, the LJ parameters being chosen such that gratifying agreement with the experimental phase diagram near the critical point is obtained. It is shown that the model gives also a reasonable account of the pair correlation function, although in the nearest neighbour shell some systematic discrepancies between the model predictions and results from simulations of fully atomistic models and experiments are found. By comparison with results from an equivalent model but with the full angle-dependent quadrupolar interaction these discrepancies are traced back to the effect of orientational correlations and of the insufficient representation of mol...

The equilibrium and flow properties of a polymer liquid confined in a brush-coated channel are studied by molecular dynamics simulations using a dissipative particle dynamics (DPD) thermostat. We focus on the regime of high-grafting density, where the brush progressively becomes a stiff and smooth, soft surface and layering of the polymer melt at the brush−melt interface is observed. We use the Gibbs criterion to localize the brush−melt interface and analyze its equilibrium fluctuation in terms of a capillary wave Hamiltonian augmented by an elastic term that accounts for the deformability of the brush. Poiseuille and Couette flows are investigated, and the slip length and location of the hydrodynamic boundary are computed. In the high-grafting regime, the brush roughness decreases and slippage is observed. The results are compared to the effective channel width, which is defined via the integrated flow rate for Poiseuille flow. Evidence of local changes of the near-surface viscosity is provided, and the ...

The properties of a single semiflexible chain tethered to a planar surface with a long-ranged attractive potential are studied by means of Monte Carlo simulations. We employ the bond fluctuation lattice model and the Wang-Landau sampling technique. We present the diagram of states for semiflexible chains consisting of N = 64 and 128 monomer units as a function of temperature T and strength of the adsorption potential, epsilon(w), and also compare this with the diagram of states for flexible chains of these two lengths. The diagram of states consists of the regions of a coil, liquid globule, solid isotropic globule, adsorbed coil, and quasi-two-dimensional solid globule with nematic bond order (or quasi-two-dimensional isotropic crystal in the case of a flexible chain). The diagram of states for the flexible case agrees with a result for the same model and a short-ranged attraction to the wall for small values of the attraction strength. For larger values, the transition to the two-dimensional behavior is more gradual in the long-ranged attractive potential. For the range of chain stiffness that we studied, the low-temperature phase in the bulk is an isotropic globule (i.e., a globule without bond order). At the attractive surface, a nematic-type bond order is induced upon adsorption, leading to ordered structures, which can be considered a precursor to the crystalline lamellae forming in real polymers. Other than that, the diagram of states in general is not changed by introducing stiffness of the chains.

When fluids are confined in slit pores between parallel walls, their static structures and their dynamical properties exhibit inhomogeneity in the z-direction perpendicular to the wall. Of particular interest are local bulk viscosity η b (z) and shear viscosity η s (z). Here, we discuss an algorithm to estimate these quantities from Green-Kubo relations using equilibrium molecular dynamics. As an application example, a polymer brush (macromolecules end-grafted to a substrate at z= 0) interacting with a solvent formed from point-like particles is given.

We present an efficient method for direct determination of the excess free energy Δ F of a nanoparticle inserted into a polymer brush. In contrast to Widom's insertion method, the present approach can be efficiently implemented by Monte Carlo or Molecular Dynamics methods also in a dense environment. In the present investigation the method is used to determine the free energy penalty Δ F ( R , D ) for placing a spherical particle with an arbitrary radius R at different positions D between the grafting plane and the brush surface. Deep inside the brush, or for dense brushes, one finds Δ F ∝ R 3 whereas for shallow nanoclusions Δ F ∝ R 2 , regardless of the particle interaction (attractive/repulsive) with the polymer. The pressure and density fields around spherical nanoinclusions in a polymer brush are also investigated. Extensive Monte Carlo simulations show that the force, exerted on the particle by the surrounding brush, depends essentially on the proximity of the nanocolloid particle to the brush surface not only in strength but also with respect to its angular distribution. For shallow nanoinclusions close to the brush surface this angular distribution is shown to result in a growing buoyant force while deep inside the brush this effect is negligible.

This review discusses Monte Carlo simulations of simple models for fluids and binary mixtures (lattice gas, lattice models for binary polymer blends, Asakura-Oosawa model of colloid-polymer mixtures) in confined geometries. The geometries considered are thin films confined by either symmetric or competing walls, as well as wedges and pyramid-shaped cones, including “closed” geometries such as double wedges and bipyramids. Testing of theoretical concepts on the phase behavior in these geometries is emphasized: There is a delicate interplay between wetting phenomena and phase separation into vapor and liquid (or two fluids of different composition, respectively). Capillary condensation phenomena occur in thin films with symmetric walls, and interface localization transitions occur in cases with competing walls. Concepts on wedge filling and cone filling are also examined.

We present a scaling theory and Monte Carlo (MC) simulation results for a flexible polymer chain slowly dragged by one end into a nanotube. We also describe the situation when the completely confined chain is released and gradually leaves the tube. MC simulations were performed for a self-avoiding lattice model with a biased chain growth algorithm, the pruned-enriched Rosenbluth method. The nanotube is a long channel opened at one end and its diameter $D$ is much smaller than the size of the polymer coil in solution. We analyze the following characteristics as functions of the chain end position $x$ inside the tube: the free energy of confinement, the average end-to-end distance, the average number of imprisoned monomers, and the average stretching of the confined part of the chain for various values of $D$ and for the number of monomers in the chain, $N$. We show that when the chain end is dragged by a certain critical distance $x^*$ into the tube, the polymer undergoes a first-order phase transition whereby the remaining free tail is abruptly sucked into the tube. This is accompanied by jumps in the average size, the number of imprisoned segments, and in the average stretching parameter. The critical distance scales as $x^*\sim ND^{1-1/\nu}$. The transition takes place when approximately 3/4 of the chain units are dragged into the tube. The theory presented is based on constructing the Landau free energy as a function of an order parameter that provides a complete description of equilibrium and metastable states. We argue that if the trapped chain is released with all monomers allowed to fluctuate, the reverse process in which the chain leaves the confinement occurs smoothly without any jumps. Finally, we apply the theory to estimate the lifetime of confined DNA in metastable states in nanotubes., Comment: 13pages, 14figures

We present simulation results for the phase behavior of a single chain for a flexible lattice polymer model using the Wang–Landau sampling idea. Using the micro-canonical density of states obtained with this method we will discuss the ability of an analysis in the micro-canonical ensemble to locate the coil-globule (continuous) and liquid–solid (first-order) transitions found for this problem using a canonical analysis.

In this article we consider mixtures of non-adsorbing polymers and rod-like colloids in the isotropic phase, which upon the addition of polymers show an effective attraction via depletion forces. Above a certain concentration, the depletant causes phase separation of the mixture. We performed Monte Carlo simulations to estimate the phase boundaries of isotropic–isotropic coexistence. To determine the phase boundaries we simulated in the grand canonical ensemble using successive umbrella sampling [J. Chem. Phys. 120 (2004) 10925]. The location of the critical point was estimated by a finite size scaling analysis. In order to equilibrate the system efficiently, we used a cluster move in which rods and spheres were exchanged.

Using Monte Carlo simulations of the bond fluctuation model, we study the formation of micelles in a confined mixture of asymmetric AB-diblock copolymers and homopolymers. The composition of the sphere-forming AB-diblock copolymers is fA = 1/8. The mixture is confined into a thin film. The film surfaces attract the minority component of the diblock with strength, eW. To efficiently sample the micelle size distribution and establish equilibrium between the surface and the bulk, we work in the semigrandcanonical ensemble, i.e. at fixed density and fixed chemical potential difference between the two types of chains, choosing a large incompatibility χN ≃ 100 (strong segregation regime). The composition of the mixture is controlled by the chemical potential difference, δµ = µcop − µhom, between copolymers and homopolymers. We study the morphology as a function of the surface interaction, eW, and the chemical potential, δµ. Only in a limited regime of parameters—i.e., in the vicinity of the adsorption transitio...

A brief review is given of methodological advances made during the past decade with the Monte Carlo sampling of equilibrium properties of simple lattice models of polymer systems, and representative applications of these new algorithms are summarized. These algorithms include Wang−Landau (WL) sampling, the pruned-enriched Rosenbluth method (PERM), and topology violating dynamic Monte Carlo algorithms such as combinations of local moves, slithering snake moves, and “double bridging” moves for the bond fluctuation model. The applications mentioned concern phase-transition-like phenomena of single chains (collapse and crystallization in bad solvents; interplay of collapse and adsorption; escape of chains from confining tubes; microphase separation in binary bottle brushes) and nontrivial correlation effects in dense systems of many chains (intramolecular long-range correlations in melts; fluctuation effects of micelles in homopolymer−block copolymer blends.) While all these examples refer to work on lattice ...

The interplay between surface and interface effects on binary AB mixtures that are confined in unconventional geometries is investigated by Monte Carlo simulations and phenomenological considerations. Both double-wedge and bi-pyramid confinements are considered and competing surface fields are applied at the two opposing halves of the system. Below the bulk critical temperature, domains of opposite order parameter are stabilized at the corresponding corners and an interface runs across the middle of the bi-partite geometry. Upon decreasing the temperature further one encounters a phase transition at which the AB symmetry is broken. The interface is localized in one of the two wedges or pyramids, respectively, and the order parameter is finite. In both cases, the transition becomes discontinuous in the thermodynamic limit but it is not a first-order phase transition. In an antisymmetric double wedge geometry the transition is closely related to the wedge-filling transition. Choosing the ratio of the cross-section L × L of the wedge and its length L y according to L y /L 3 = const., simulations and phenomenological consideration show that the new type of phase transition is characterized by critical exponents α = 3/4, β = 0, and γ = 5/4 for the specific heat, order parameter, and susceptibility, respectively. In an antisymmetric bi-pyramid the transition occurs at the cone-filling transition of a single pyramid. The important critical fluctuations are associated with the uniform translation of the interface and they can be described by a Landau-type free energy. Monte Carlo results provide evidence that the coefficients of this Landau-type free energy exhibit a system-size dependence, which gives rise to critical amplitudes that diverge with system size and result in a transition that becomes discontinuous in the thermodynamic limit.

The concept of double-well thermodynamic potentials, ubiquitous since the van der Waals description of the vapour-to-liquid transition and the Landau theory of phase transitions, is critically re-examined. Particular emphasis is put on the extent to which spinodal curves (separating ‘metastable’ from ‘unstable’ states) are meaningful. It is argued that in full thermodynamic equilibrium spinodals are well-defined when one either considers finite subsystems of an infinitely large system, or systems with all linear dimensions finite. Evidence is given that in a finite (cubic) d-dimensional box the spinodals correspond (in a fluid) to the rounded ‘droplet evaporation’ or ‘bubble condensation’ transitions, respectively, and are shifted inside the bulk coexistence region by amounts of order L − d / (d + 1), for a box of linear dimension L. The region in between the spinodals is the finite size analogue of the bulk two-phase coexistence region. Consequences for coarse-grained descriptions of the kinetics of phas...

The properties of Ising square lattices with nearest neighbor ferromagnetic exchange confined in a corner geometry, are studied by means of Monte Carlo simulations. Free boundary conditions at which boundary magnetic fields ± h are applied, i.e., at the two boundary rows ending at the lower left corner a field + h acts, while at the two boundary rows ending at the upper right corner a field − h acts. For temperatures T less than the critical temperature T c of the bulk, this boundary condition leads to the formation of two domains with opposite orientation of the magnetization direction, separated by an interface which for T larger than the filling transition temperature T f ( h ) runs from the upper left corner to the lower right corner, while for T T f ( h ) this interface is localized either close to the lower left corner or close to the upper right corner. It is shown that for T = T f ( h ) the magnetization profile m ( z ) in the z-direction normal to the interface simply is linear and the interfacial width scales as w ∝ L , while for T > T f ( h ) it scales as w ∝ L . The distribution P ( l ) of the interface position l (measured along the z-direction from the corners) decays exponentially for T T f ( h ) from either corner, is essentially flat for T = T f ( h ) , and is a Gaussian centered at the middle of the diagonal for T > T f ( h ) . Unlike the findings for critical wetting in the thin film geometry of the Ising model, the Monte Carlo results for corner wetting are in very good agreement with the theoretical predictions.

When simple or complex fluids are confined to ultrathin films or channels or other cavities of nanoscopic linear dimensions, the interplay of finite size and surface controls the phase behavior, and may lead to phase transitions rather different from the corresponding phenomena in the bulk. Monte Carlo simulation is a very suitable tool to clarify the complex behavior of such systems, since the boundary conditions providing the confinement can be controlled and arbitrarily varied, and detailed structural information on the inhomogeneous states of the considered systems is available. Examples used to illustrate these concepts include simple Ising models in pores and double-pyramid-shaped cavities with competing surface fields, where novel types of interface localization–delocalization phenomena occur accompanied by “macroscopic” fluctuations, and colloid-polymer mixtures confined in slit pores. Finite size scaling concepts are shown to be a useful tool also for such systems “in between” the dimensionalities.

The scaling concepts for isolated flexible macromolecules in good solvent grafted with one chain end to a flat surface (polymer mushrooms) as well as for layers of many overlapping end-grafted chain molecules (polymer brushes) are introduced. Monte Carlo attempts to test these concepts are briefly reviewed. Then the extension of these concepts to polymer brushes grafted to the interior of a cylinder surface is discussed. Molecular Dynamics results on chain average linear dimensions in the direction normal to the grafting surface and in axial direction are described, as well as distribution functions for the density of end monomers and of all monomers of the chains. It is argued that under typical conditions reachable in either simulation or experiment the data fall in a crossover regime, where no simple power-laws (as derived from the scaling description) hold. Moreover, extensions of the Daoud-Cotton blob picture to the cylinder geometry, implying that each chain in such a cylindrical brush is confined into a conical sector, are invalid; thus, chain ends in densely filled cylinders are not restricted to stay in the hemicylinder containing the grafting site of the chain.

We analyze the collapse transition of flexible homopolymer chains in the bond-fluctuation model employing the Wang-Landau Monte Carlo algorithm. The coil-globule transition is followed by a first order transition into a solid state occurring in the collapsed globule. In the thermodynamic limit (chain length to infinity) the topology of the phase diagram depends on the range of the attractive interaction between the monomers. For sufficiently large interaction range a normal behaviour of a continuous coil-globule transition at the Θ-temperature followed by a crystallization transition at lower temperature is observed. For short interaction range the first-order transition asymptotically can pre-empt the coil-globule transition such that a first-order collapse from a random-coil into a crystalline globule occurs. This vanishing of the liquid globule state is similar to the vanishing of the vapour-liquid coexistence found for colloidal suspensions interacting by a short-ranged attraction.

The structure of bottle-brush polymers with a rigid backbone and flexible side chains is studied in three dimensions, varying the grafting density, the side chain length, and the solvent quality. Some preliminary results of theoretical scaling considerations for one-component bottle-brush polymers in a good solvent are compared with Monte Carlo simulations of a simple lattice model. For the simulations a variant of the pruned-enriched Rosenbluth method (PERM) allowing for simultaneous growth of all side chains in the Monte Carlo sampling is employed. For a symmetrical binary (A,B) bottle-brush polymer, where two types (A,B) of flexible side chains are grafted with one chain end to the backbone in an alternating way, varying repulsive binary interactions between unlike monomers and the solvent quality, it is found that phase separation into an A-rich part of the cylindrical molecule and a B-rich part can occur only locally. Long range order (in the direction of the backbone) does not occur, and hence the transition from the randomly mixed state of the bottle-brush to the phase-separated structure is strongly rounded, in contrast to the corresponding mean field predictions of a sharp transition to a Janus cylinder phase-separated structure. This lack of a phase transition can be understood from an analogy with spin models in one dimension. By estimating the correlation length for this phase separation along the backbone as a function of side chain length and solvent quality, we present strong evidence that no sharp phase transition occurs.

solids to complex multi-component materials and even biological matter, are governed by well understood laws of physics: on the relevant scales of length and time; the appropriate description would be just the Schrodinger equation for the quantum-many-body problem of the nuclei and electrons interacting with Coulomb forces. Statistical mechanics would then provide the framework to extend this quantum theory of condensed matter towards a statistical description in terms of averages taken at nonzero temperature. However, this program cannot be carried out straightforwardly: even dealing only with the associated electrons is still a challenge for the methods of quantum chemistry. Similarly, standard statistical mechanics makes precise explicit statements only on the properties of systems for which the manybody problem can be effectively reduced to a problem of independent particles or quasi-particles. Such problems are, for instance, ideal gases, paramagnets, or the multidimensional harmonic oscillator describing phonons in harmonic crystals. While all these problems are useful and educative (we hence teach them all to students to illustrate the spirit of the general theoretical framework) they do not encompass most of the problems of interest in condensed matter physics: the interactions among the considered degrees of freedom introduce nontrivial correlations between them. Systematic perturbation-theory type methods usually do not lead very far, and uncontrolled closed form approximations often fail utterly. A well-known example is the study of the liquid gas transition in fluids: simple theories such as the van der Waals equation and its extensions (e.g. the "perturbed chain" statistical associating fluid theory [PC-SAFT] intended to model the equation of state of polymer melts [1]) give rise to spurious loops in the isotherms, fail to describe the critical behavior correctly, and may even predict completely unphysical phase equilibria that do not correspond to any physical behavior of the system but are mere artefacts of the inappropriate approximations [1]. The theory of condensed matter systems contains a never-ending list of such failures! Hence, until about 50 years ago, condensed matter theory suffered from the basic problem that only a formal framework for the theoretical description in terms of quantum theory and statistical mechanics did exist, while a reliable set of tools that would allow accurate explicit predictions to be made for the static and dynamic properties of these systems from first principles was simply missing! This unsatisfactory situation has changed fundamentally through the invention of computer simulation, which provides a much more promising approach to these problems. Computer simulation is a novel methodic route allowing the Condensed matter theory by computer simulation: from materials to chemical biology [DOI: 10.1051/EPN:2007009]

A two-dimensional magnetic Ising system confined in an L × D geometry ( L ⪡ D ) in the presence of competing magnetic fields (h) acting at opposite walls along the D -direction, exhibits an interface between domains of different orientation that run parallel to the walls. In the limit L → ∞ , this interface undergoes a wetting transition that occurs at the critical curve T w ( h ) , so that for T T w ( h ) such an interface is bound to the walls, while for T w ( h ) ⩽ T T cb the interface is freely fluctuating around the center of the film, where T cb is the bulk critical temperature. By considering both short- and long-range magnetic fields acting at the walls, we study the divergence of the (equilibrated) average position of the interface when approaching the wetting critical point. Furthermore, starting from a monodomain structure with the interface bound to one wall, we also study the dynamics of the interface unbinding.

When binary mixtures are confined into nanoscopic slit pores, an intricate interplay between surface enrichment (wetting) of one component and lateral phase separation occurs. After a brief review of the static equilibrium phase diagram of such systems, a discussion of the kinetics of phase separation is given. Considering quenches from an initially homogeneous distribution of the two species in the slit, it is shown by molecular dynamics simulation that typically in the initial stages a stratified structure develops, with enrichment layers of the preferred component at the walls of the slit pore. Then this laterally homogeneous structure breaks up into domains, which coarsen with time according to a power law with a 2/3 exponent. This growth law must be attributed to a hydrodynamic mechanism, since corresponding simulations of a diffusive Ginzburg-Landau model yield an exponent of 1/3 only. The relation to spinodal decomposition in d=2 space dimensions is briefly discussed.

The article conbtains sections titled: 1 Introduction and Overview 2 Random-Number Generation 2.1 General Introduction 2.2 Properties That a Random-Number Generator (RNG) Should Have 2.3 Comments about a Few Frequently Used Generators 3 Simple Sampling of Probability Distributions Using Random Numbers 3.1 Numerical Estimation of Known Probability Distributions 3.2 “Importance Sampling” versus “Simple Sampling” 3.3 Monte-Carlo as a Method of Integration 3.4 Infinite Integration Space 3.5 Random Selection of Lattice Sites 3.6 The Self-Avoiding Walk Problem 3.7 Simple Sampling versus Biased Sampling: the Example of SAWs Continued 4 Survey of Applications to Simulation of Transport Processes 4.1 The “Shielding Problem” 4.2 Diffusion-Limited Aggregation (DLA) 5 Monte-Carlo Methods in Statistical Thermodynamics: Importance Sampling 5.1 The General Idea of the Metropolis Importance-Sampling Method 5.2 Comments on the Formulation of a Monte-Carlo Algorithm 5.3 The Dynamic Interpretation of the Monte-Carlo Method 5.4 Monte-Carlo Study of the Dynamics of Fluctuations Near Equilibrium and of the Approach toward Equilibrium 5.5 The Choice of Statistical Ensembles 6 Accuracy Problems: Finite-Size Problems, Dynamic Correlation of Errors, Boundary Conditions 6.1 Finite-Size–Induced Rounding and Shifting of Phase Transitions 6.2 Different Boundary Conditions: Simulation of Surfaces and Interfaces 6.3 Estimation of Statistical Errors 7 Sampling of Free Energies and Free Energy Barriers 7.1 Bulk Free Energies 7.2 Interfacial Free Energies 7.3 Transition Path Sampling 8 Quantum Monte-Carlo Techniques 8.1 General Remarks 8.2 Path-Integral Monte-Carlo Methods 8.3 A Classical Application: the Momentum Distribution of Fluid He 8.4 A Few Qualitative Comments on Fermion Problems 9 Lattice Gauge Theory 9.1 Some Basic Ideas of Lattice Gauge Theory 9.2 A Famous Application 10 Selected Applications in Classical Statistical Mechanics of Condensed Matter 10.1 Metallurgy and Materials Science 10.2 Polymer Science 10.3 Surface Physics 11 Concluding Remarks Glossary Keywords: self-averaging; attrition problem; sweep step; statistical inefficiency; quenched approximation; random-number generator

Semiflexible polymers end-grafted to a repulsive planar substrate under good solvent conditions are studied by scaling arguments, computer simulations, and self-consistent field theory. Varying the chain length N, persistence length lp, and grafting density σg, the chain linear dimensions and distribution functions of all monomers and of the free chain ends are studied. Particular attention is paid to the limit of very small σg, where the grafted chains behave as "mushrooms" no longer interacting with each other. Unlike a flexible mushroom, which has a self-similar structure from the size (a) of an effective monomer up to the mushroom height (h/a ∝ N(v), ν ≈ 3/5), a semiflexible mushroom (like a free semiflexible chain) exhibits three different scaling regimes, h/a ∝ N for contour length L = Nalp, a Gaussian regime, h/a ∝ (Llp)(1/2)/a for lp ≪ L ≪ R* ∝ (lp(2)/a), and a regime controlled by excluded volume, h/a ∝ (lp/a)(1/5)N(ν). The semiflexible brush is predicted to scale as h/a ∝ (lpaσg)(1/3)N in the excluded volume regime, and h/a ∝ (lpa(3)σ(2))(1/4)N in the Gaussian regime. Since in the volume taken by a semiflexible mushroom excluded-volume interactions are much weaker in comparison to a flexible mushroom, there occurs an additional regime where semiflexible mushrooms overlap without significant chain stretching. Moreover, since the size of a semiflexible mushroom is much larger than the size of a flexible mushroom with the same N, the crossover from mushroom to brush behavior is predicted to take place at much smaller densities than for fully flexible chains. The numerical results, however, confirm the scaling predictions only qualitatively; for chain lengths that are relevant for experiments, often intermediate effective exponents are observed due to extended crossovers.

As a model for a suspension of hard-sphere like colloidal particles where small nonadsorbing dissolved polymers create a depletion attraction, we introduce an effective colloid-colloid potential closely related to the Asakura-Oosawa model but that does not have any discontinuities. In simulations, this model straightforwardly allows the calculation of the pressure from the Virial formula, and the phase transition in the bulk from the liquid to crystalline solid can be accurately located from a study where a stable coexistence of a crystalline slab with a surrounding liquid phase occurs. For this model, crystalline nuclei surrounded by fluid are studied both by identifying the crystal-fluid interface on the particle level (using suitable bond orientational order parameters to distinguish the phases) and by "thermodynamic" means. I.e., the latter method amounts to compute the enhancement of chemical potential and pressure relative to their coexistence values. We show that the chemical potential can be obtained from simulating thick films, where one wall with a rather long range repulsion is present, since near this wall the Widom particle insertion method works, exploiting the fact that the chemical potential in the system is homogeneous. Finally, the surface excess free energy of the nucleus is obtained, for a wide range of nuclei volumes. From this method, it is established that classical nucleation theory works, showing that for the present model the anisotropy of the interface excess free energy of crystals and their resulting nonspherical shape has only a very small effect on the barrier.

Using Monte Carlo simulations of the bond fluctuation model and self-consistent field calculations, we study the formation of micelles in a mixture of homopolymers and asymmetric AB-diblock copolymers with composition, fA = 1/8. Both types of molecules are fully flexible and have identical length. We work in the semi-grand-canonical ensemble, i.e., we fix the monomer density and incompatibility, χN ≃ 100 (strong segregation regime), and control the composition of the mixture via the exchange chemical potential, δμ ≡ μAB − μB between the copolymers and homopolymers. The Monte Carlo simulation comprises moves that allow homopolymers to mutate into AB-diblock copolymers and vice versa. These moves are very efficient in equilibrating the configurations. We accurately locate the critical micelle concentration, study the micellar size distribution and characterize the shape of the micelles by the tensor of gyration and radial density profiles. The results of the simulations are quantitatively compared to predic...

Confining a binary mixture, one can profoundly alter its miscibility behavior. The qualitative features of miscibility in confined geometry are ratheruniversal and shared by polymer mixtures as well as small molecules, but the unmixing transition in the bulk and the wetting transition are typically well separated in polymer blends. The interplay between wetting and miscibility of a symmetric polymer mixture via large-scale Monte Carlo simulations in the framework of the bond fluctuation model and via numerical self–consistent field calculations is studied. The film surfaces interact with the monomers via short ranged potentials, and the wetting transition of the semi–infinite system is of first order. It can be accurately located in the simulations by measuring the surface and interface tensions and using Young’s equation. If both surfaces in a film attract the same component, capillary condensation occurs and the critical point is close to the critical point of the bulk. If surfaces attract different components, an interface localization/delocalization occurs which gives rise to phase diagrams with two critical points in the vicinity of the pre-wetting critical point of the semi–infinite system. The crossover between these two types of phase diagrams as a function of the surface field asymmetry is studied. The dependence of the phase diagram on the film thickness Δ for antisymmetric surface fields is investigated. Upon decreasing the film thickness, the two critical points approach the symmetry axis of the phase diagram, and below a certain thickness Δtri, there remains only a single critical point at the symmetric composition. This corresponds to a second-order interface localization/delocalization transition even though the wetting transition is of first order. At a specific film thickness, Δtri, tricritical behavior is found. The behavior of antisymmetric films is compared with the phase behavior in an antisymmetric double wedge. While the former is the analog of the wetting transition of a planar surface, the latter is related to the filling behavior of a single wedge. Evidence for a second-order interface localization/delocalization transition in an antisymmetric double wedge is presented, and its unconventional critical behavior is related to the predictions of Parry et al. (Phys. Rev. Lett. 83:5535 (1999)) for wedge filling. The critical behavior differs from the Ising universality class and is characterized by strong anisotropic fluctuations.

The behaviour of a flexible polymer chain in solvents of variable quality in dilute solution is discussed both in the bulk and in the presence of an adsorbing wall. Monte Carlo simulations of coarse-grained bead-spring models and of the bond fluctuation model are presented and interpreted in terms of phenomenological theories and scaling concepts. Particular attention is paid to the behaviour of the polymer chain when the temperature of the polymer solution gets lower than the Theta temperature. It is argued that the adsorption transition line at the Theta temperature splits into lines of wetting and drying transitions of polymer globules attached to the wall. In addition, it is shown that the coil-globule transition is followed by a second transition, which in the bond fluctuation model is a crystallization into a regular lattice structure. Performing a finite size scaling analysis on the two transitions it is shown that (for the chosen model) both transitions coincide in the thermodynamic limit, corresponding to a direct collapse of the random coil into the crystal without intermediate coil-globule transition. The implications of this result for the standard mean field or tricritical theory of the coil-globule transition based on a truncated virial expansion are discussed.

In this work we report on molecular dynamics simulations of the embedding process of a nano-particle into a polymeric film as a function of temperature. This process has been employed experimentally in recent years to test for a shift of the glass transition of a material due to the confined film geometry and to test for the existence of a liquid-like layer on top of a glassy polymer film. The embedding process is governed thermodynamically by the prewetting properties of the polymer on the nano-particle. We show that the dynamics of the process depends on the Brownian motion characteristics of the nano-particle in and on the polymer film. It displays large sample to sample variations, suggesting that it is an activated process. On the timescales of the simulation an embedding of the nano-particle is only observed for temperatures above the bulk glass transition temperature of the polymer, agreeing with experimental observations on noble metal clusters of comparable size.

A two-dimensional magnetic Ising system confined in an L × D geometry () in the presence of competing magnetic fields (h) acting at opposite walls along the D-direction exhibits an interface between domains of different orientation that runs parallel to the walls. In the limit of infinite film thickness () this interface undergoes a wetting transition that occurs at the critical curve Tw(h), so that for T

Coarse-graining of chemical structure of macromolecules in the melt is investigated using extensive molecular dynamics simulation data which are based on a united atom force-field model of polybutadiene. Systematically increasing the number, n, of the united atoms approximated by an effective coarse-grained monomer, we study the influence of degree of coarse-graining on the structure functions such as the segment-segment intermolecular and intramolecular correlation functions. These results are compared to Monte Carlo simulations of the corresponding coarse-grained bead-spring model and Chen-Kreglewski potential for chain molecules. In contrast to the atomistic chemically realistic model of polybutadiene, the bending and torsional potentials are not included into the coarse-grained models. Nevertheless, for a range of intermediate values of n a good qualitative agreement between intra- and intermolecular coarse-grained correlations of the atomistic model and the coarse-grained bead-spring model is found on large and intermediate length scales, but deviations occur on length scales well below one nanometer. The structure functions obtained for the Chen-Kreglewski chains exhibit many artificial features.

The interfacial tension between coexisting phases of a material is an important parameter in the description of many phenomena such as crystallization, and even today its accurate measurement remains difficult. We have studied logarithmic finite-size corrections in the determination of the interfacial tension with large scale Monte Carlo simulations, and have identified several novel contributions which not only depend on the ensemble, but also on the type of the applied boundary conditions. We present results for the Lennard-Jones system and the Ising model, as well as for hard spheres, which are particularly challenging. In the future, these findings will contribute to the understanding and determination of highly accurate interfacial properties with computer simulations, and will be used in the study of nucleation of colloidal crystals. As a first application, we compare the Laplace pressure of a crystalline nucleus surrounded by liquid as obtained from simulations with classical nucleation theory.

Introduction In most of the discussion presented so far in this book, the quantum character of atoms and electrons has been ignored. The Ising spin models have been an exception, but since the Ising Hamiltonian is diagonal (in the absence of a transverse magnetic field), all energy eigenvalues are known and the Monte Carlo sampling can be carried out just as in the case of classical statistical mechanics. Furthermore, the physical properties are in accord with the third law of thermodynamics for Ising-type Hamiltonians (e.g. entropy S and specific heat vanish for temperature T → 0, etc.) in contrast to the other truly classical models dealt with in previous chapters (e.g. classical Heisenberg spin models, classical fluids and solids, etc.) which have many unphysical low temperature properties. A case in point is a classical solid for which the specific heat follows the Dulong–Petit law, C = 3 Nk B , as T → 0, and the entropy has unphysical behavior since S → –∞. Also, thermal expansion coefficients tend to non-vanishing constants for T → 0 while the third law implies that they must be zero. While the position and momentum of a particle can be specified precisely in classical mechanics, and hence the groundstate of a solid is a perfectly rigid crystal lattice (motionless particles localized at the lattice points), in reality the Heisenberg uncertainty principle forbids such a perfect rigid crystal, even at T → 0, due to zero point motions which ‘smear out’ the particles over some region around these lattice points. This delocalization of quantum-mechanical particles increases as the atomic mass is reduced; therefore, these quantum effects are most pronounced for light atoms like hydrogen in metals, or liquid helium. Spectacular phenomena like superfluidity are a consequence of the quantum nature of the particles and have no classical counterpart at all. Even for heavier atoms, which do not show superfluidity because the fluid–solid transition intervenes before a transition from normal fluid to superfluid could occur, there are genuine effects of quantum nature. Examples include the isotope effects (remember that in classical statistical mechanics the kinetic energy part of the Boltzmann factor cancels out from all averages, and thus in thermal equilibrium no property depends explicitly on the mass of the particles).

Dealing with all aspects of Monte Carlo simulation of complex physical systems encountered in condensed-matter physics and statistical mechanics, this book provides an introduction to computer simulations in physics. This fourth edition contains extensive new material describing numerous powerful algorithms not covered in previous editions, in some cases representing new developments that have only recently appeared. Older methodologies whose impact was previously unclear or unappreciated are also introduced, in addition to many small revisions that bring the text and cited literature up to date. This edition also introduces the use of petascale computing facilities in the Monte Carlo arena. Throughout the book there are many applications, examples, recipes, case studies, and exercises to help the reader understand the material. It is ideal for graduate students and researchers, both in academia and industry, who want to learn techniques that have become a third tool of physical science, complementing experiment and analytical theory.