Phase behavior and dynamics in a colloid-polymer mixture under spherical confinement.

From studies via molecular dynamics simulations, we report results on structure and dynamics in mixtures of active colloids and passive polymers that are confined inside a spherical container with a repulsive boundary. All interactions in the fully passive limit are chosen in such a way that in equilibrium coexistence between colloid-rich and polymer-rich phases occurs. For most part of the studies the chosen compositions give rise to Janus-like structure: nearly one side of the sphere is occupied by the colloids and the rest by the polymers. This partially wet situation mimics approximately a neutral wall in the fully passive scenario. Following the introduction of a velocity-aligning activity to the colloids, the shape of the polymer-rich domain changes to that of an ellipsoid, around the long axis of which the colloid-rich domain attains a macroscopic angular momentum. In the steady state, the orientation of this axis evolves via diffusion, magnitude of which depends upon the strength of activity, but only weakly.

Surface enrichment and interdiffusion in blends of semiflexible polymers of different stiffness.

A model for a mixture of two kinds of semiflexible polymers (A and B) with the same chain length (NA = NB = 32), but different persistence lengths, confined between parallel planar repulsive walls in a common good solvent is studied by molecular dynamics simulations. In the isotropic phase at low polymer concentrations, both polymers are repelled by the walls, and the system is anisotropic near the walls over a range controlled by the polymer linear dimensions. Close to the concentrations where in the bulk nematic order sets in, precursors of thick nematic layers at the walls are observed, strongly enriched by a stiffer component, which hence is depleted in the center of the slit pore. At larger concentrations, where in the bulk a uniformly mixed nematic phase occurs, the enrichment of B-chains at the walls is rather minor, extending over the scale of the transverse correlation length of concentration fluctuations, which is of the order of a few monomeric diameters only for the present model. In this ordered phase, both self-diffusion and interdiffusion of chains (in the direction perpendicular to the director) are found to be significantly slowed down in comparison to dilute solutions. Since equilibration times scale with the square of the slit thickness, incomplete equilibration is predicted when polymeric coatings on substrate containing polymers differing in stiffness are produced.

Wetting transitions of polymer solutions: Effects of chain length and chain stiffness.

Wetting and drying phenomena are studied for flexible and semiflexible polymer solutions via coarse-grained molecular dynamics simulations and density functional theory calculations. This study is based on the use of Young's equation for the contact angle, determining all relevant surface tensions from the anisotropy of the pressure tensor. The solvent quality (or effective temperature, equivalently) is varied systematically, while all other interactions remain unaltered. For flexible polymers, the wetting transition temperature Tw increases monotonically with chain length N, while the contact angle at temperatures far below Tw is independent of N. For semiflexible polymer solutions, Tw varies non-monotonically with the persistence length: Initially, Tw increases with increasing chain stiffness and reaches a maximum, but then a sudden drop of Tw is observed, which is associated with the isotropic-nematic transition of the system.

Cylindrical confinement of solutions containing semiflexible macromolecules: surface-induced nematic order versus phase separation.

Solutions of semiflexible polymers confined in cylindrical pores with repulsive walls are studied by Molecular Dynamics simulations for a wide range of polymer concentrations. Both the case where both lengths are of the same order and the case when the persistence length by far exceeds the contour length are considered, and the enhancement of nematic order along the cylinder axis is characterized. With increasing density the character of the surface effect changes from depletion to the formation of a layered structure. For binary 50 : 50 mixtures of the two types of polymers an interplay between surface enrichment of the stiffer component and the isotropic-nematic transition is found, and a phase separated structure with cylindrical symmetry occurs, with the isotropic phase located around the cylinder axis. For melt densities the mixed nematic phase forms at the wall a layer with a screw-like structure of a tilted smectic phase. The observed behavior is tentatively interpreted in terms of the competition of the chain orientational entropy with entropy of mixing and excluded volume due to the wall.

When does Wenzel's extension of Young's equation for the contact angle of droplets apply? A density functional study.

The contact angle of a liquid droplet on a surface under partial wetting conditions differs for a nanoscopically rough or periodically corrugated surface from its value for a perfectly flat surface. Wenzel's relation attributes this difference simply to the geometric magnification of the surface area (by a factor rw), but the validity of this idea is controversial. We elucidate this problem by model calculations for a sinusoidal corrugation of the form zwall(y) = Δ cos(2πy/λ), for a potential of short range σw acting from the wall on the fluid particles. When the vapor phase is an ideal gas, the change in the wall-vapor surface tension can be computed exactly, and corrections to Wenzel's equation are typically of the order σwΔ/λ2. For fixed rw and fixed σw, the approach to Wenzel's result with increasing λ may be nonmonotonic and this limit often is only reached for λ/σw > 30. For a non-additive binary mixture, density functional theory is used to work out the density profiles of both coexisting phases for planar and corrugated walls as well as the corresponding surface tensions. Again, deviations from Wenzel's results of similar magnitude as in the above ideal gas case are predicted. Finally, a crudely simplified description based on the interface Hamiltonian concept is used to interpret the corresponding simulation results along similar lines. Wenzel's approach is found to generally hold when λ/σw ≫ 1 and Δ/λ < 1 and under conditions avoiding proximity of wetting or filling transitions.

Linear Dimensions of Adsorbed Semiflexible Polymers: What Can Be Learned about Their Persistence Length?

Conformations of partially or fully adsorbed semiflexible polymer chains are studied varying both contour length L, chain stiffness, κ, and the strength of the adsorption potential over a wide range. Molecular dynamics simulations show that partially adsorbed chains (with "tails," surface attached "trains," and "loops") are not described by the Kratky-Porod wormlike chain model. The crossover of the persistence length from its three-dimensional value (ℓ_{p}) to the enhanced value in two dimensions (2ℓ_{p}) is analyzed, and excluded volume effects are identified for L≫ℓ_{p}. Consequences for the interpretation of experiments are suggested. We verify the prediction that the adsorption threshold scales as ℓ_{p}^{-1/3}.

Second inflection point of water surface tension in the deeply supercooled regime revealed by entropy anomaly and surface structure using molecular dynamics simulations.

The surface tension of supercooled water is of fundamental importance in physical chemistry and materials and atmospheric sciences. Controversy, however, exists over its temperature dependence in the supercooled regime, especially on the existence of the "second inflection point (SIP)". Here, we use molecular dynamics simulations of the SPC/E water model to study the surface tension of water (σw) as a function of temperature down to 198.15 K, and find a minimum point of surface excess entropy per unit area around ∼240-250 K. Additional simulations with the TIP4P/2005 water model also show consistent results. Hence, we predict an SIP of σw roughly in this region, at the boundary where the "no man's land" happens. The increase of surface entropy with decreasing temperature in the region below the inflection point is clearly an anomalous behavior, unknown for simple liquids. Furthermore, we find that σw has a near-linear correlation with the interfacial width, which can be well explained by the capillary wave theory. Deep in the supercooled regime, a compact water layer at the interface is detected in our simulations, which may be a key component that contributes to the deviation of surface tension from the International Association for the Properties of Water and Steam relationship. Our findings may advance the understanding of the origin of the anomalous properties of liquid water in the supercooled regime.

Phase behavior of flexible and semiflexible polymers in solvents of varying quality.

The interplay of nematic order and phase separation in solutions of semiflexible polymers in solvents of variable quality is investigated by density functional theory (DFT) and molecular dynamics (MD) simulations. We studied coarse-grained models, with a bond-angle potential to control chain stiffness, for chain lengths comparable to the persistence length of the chains. We varied both the density of the monomeric units and the effective temperature that controls the quality of the implicit solvent. For very stiff chains, only a single transition from an isotropic fluid to a nematic is found, with a phase diagram of "swan-neck" topology. For less stiff chains, however, also unmixing between isotropic fluids of different concentration, ending in a critical point, occurs for temperatures above a triple point. The associated critical behavior is examined in the MD simulations and found compatible with Ising universality. Apart from this critical behavior, DFT calculations agree qualitatively with the MD simulations.

Heterogeneous nucleation of a droplet pinned at a chemically inhomogeneous substrate: A simulation study of the two-dimensional Ising case.

Heterogeneous nucleation is studied by Monte Carlo simulations and phenomenological theory, using the two-dimensional lattice gas model with suitable boundary fields. A chemical inhomogeneity of length b at one boundary favors the liquid phase, while elsewhere the vapor is favored. Switching on the bulk field Hb favoring the liquid, nucleation and growth of the liquid phase starting from the region of the chemical inhomogeneity are analyzed. Three regimes occur: for small fields, HbHb*), the droplets nucleated at the chemical inhomogeneity grow to the full system size. While the relaxation time for the growth scales as τG∝Hb-1, the nucleation time τN scales as lnτN∝Hb-1. However, the prefactor in the latter relation, as evaluated for our simulations results, is not in accord with an extension of the Volmer-Turnbull theory to two-dimensions, when the theoretical contact angle θc is used.

Nematic order in solutions of semiflexible polymers: Hairpins, elastic constants, and the nematic-smectic transition.

Coarse-grained models of lyotropic solutions of semiflexible polymers are studied by both molecular dynamics simulations and density functional theory calculations, using an implicit solvent bead-spring model with a bond-angle potential. We systematically vary the monomer density, persistence length, and contour length over a wide range and explore the full range from the isotropic-nematic transition to the nematic-smectic transition. In the nematic regime, we span the entire regime from rigid-rod like polymers to thin wormlike chains, confined in effective straight tubes caused by the collective nematic effective ordering field. We show that the distribution of bond angles relative to the director is well described by a Gaussian, irrespective of whether the chains are rod-like or rather flexible. However, the related concept of "deflection length" is shown to make sense only in the latter case for rather dilute solutions since otherwise the deflection length is of the order of about two bond lengths only. When the solution is semi-dilute, a substantial renormalization of the persistence length occurs, while this effect is absent in the isotropic phase even at rather high monomer densities. The effective radii of the "tubes" confining the chains in the related description of orientational ordering are significantly larger than the distances between neighboring chains, providing evidence for a pronounced collective character of orientational fluctuations. Hairpins can be identified close to the isotropic-nematic transition, and their probability of occurrence agrees qualitatively with the Vroege-Odijk theory. The corresponding theoretical predictions for the elastic constants, however, are not in good agreement with the simulations. We attribute the shortcomings of the theories to their neglect of the coupling between local density and orientational fluctuations. Finally, we detected for this model a transition to a smectic phase for reduced monomer densities near 0.7.

Smectic C and Nematic Phases in Strongly Adsorbed Layers of Semiflexible Polymers.

Molecular dynamics simulations of semiflexible polymers in a good solvent reveal a dense adsorbed layer when the solution is exposed to an attractive planar wall. This layer exhibits both a nematic and a smectic phase (smA for short and smC for longer chains) with bond vectors aligned strictly parallel to the wall. The tilt angle of the smC phase increases strongly with the contour length of the polymers. The isotropic-nematic transition is a Kosterlitz-Thouless transition and also the nematic-smectic transition is continuous. Our finding demonstrates thus a two-dimensional realization of different liquid crystalline phases, ubiquitous in three dimensions, that occurs in a single monomolecular layer ordered at least over mesoscopic scales.

Semiflexible Polymers in Spherical Confinement: Bipolar Orientational Order Versus Tennis Ball States.

Densely packed semiflexible polymers with contour length L confined in spheres with radius R of the same order as L cannot exhibit uniform nematic order. Depending on the chain stiffness (which we vary over a wide range), highly distorted structures form with topological defects on the sphere surface. These structures are completely different from previously observed ones of very long chains winding around the inner surface of spheres and from nematic droplets. At high densities, a thin shell of polymers close to the sphere surface exhibits a tennis ball texture due to the confinement-induced gradual bending of polymer bonds. In contrast, when the contour length of the chains is significantly smaller than the radius of the confining sphere, a few bent smectic layers form in the sphere. Molecular dynamics simulations demonstrate these complex structures, and suitable order parameters characterizing them are proposed.

Semiflexible polymers confined in a slit pore with attractive walls: two-dimensional liquid crystalline order versus capillary nematization.

Semiflexible polymers under good solvent conditions interacting with attractive planar surfaces are investigated by Molecular Dynamics (MD) simulations and classical Density Functional Theory (DFT). A bead-spring type potential complemented by a bending potential is used, allowing variation of chain stiffness from completely flexible coils to rod-like polymers whose persistence length by far exceeds their contour length. Solvent is only implicitly included, monomer-monomer interactions being purely repulsive, while two types of attractive wall-monomer interactions are considered: (i) a strongly attractive Mie-type potential, appropriate for a strictly structureless wall, and (ii) a corrugated wall formed by Lennard-Jones particles arranged on a square lattice. It is found that in dilute solutions the former case leads to the formation of a strongly adsorbed surface layer, and the profile of density and orientational order in the z-direction perpendicular to the wall is predicted by DFT in nice agreement with MD. While for very low bulk densities a Kosterlitz-Thouless type transition from the isotropic phase to a phase with power-law decay of nematic correlations is suggested to occur in the strongly adsorbed layer, for larger densities a smectic-C phase in the surface layer is detected. No "capillary nematization" effect at higher bulk densities is found in this system, unlike systems with repulsive walls. This finding is attributed to the reduction of the bulk density (in the center of the slit pore) due to polymer adsorption on the attractive wall, for a system studied in the canonical ensemble. Consequently in a system with two attractive walls nematic order in the slit pore can occur only at a higher density than for a bulk system.

Conformations and orientational ordering of semiflexible polymers in spherical confinement.

Semiflexible polymers in lyotropic solution confined inside spherical nanoscopic "containers" with repulsive walls are studied by molecular dynamics simulations and density functional theory, as a first step to model confinement effects on stiff polymers inside of miniemulsions, vesicles, and cells. It is shown that the depletion effects caused by the monomer-wall repulsion depend distinctly on the radius R of the sphere. Further, nontrivial orientational effects occur when R, the persistence length ℓp, and the contour length L of the polymers are of similar magnitude. At intermediate densities, a "shell" of wall-attached chains is forming, such that the monomers belonging to those chains are in a layer at about the distance of one monomer from the container wall. At the same time, the density of the centers of mass of these chains is peaked somewhat further inside, but still near the wall. However, the arrangement of chains is such that the total monomer density is almost uniform in the sphere, apart from a small layering peak at the wall. It is shown that excluded volume effects among the monomers are crucial to account for this behavior, although they are negligible for comparable isolated single semiflexible chains of the same length.

Estimation of the critical behavior in an active colloidal system with Vicsek-like interactions.

We study numerically the critical behavior of a modified, active Asakura-Oosawa model for colloid-polymer mixtures. The colloids are modeled as self-propelled particles with Vicsek-like interactions. This system undergoes phase separation between a colloid-rich and a polymer-rich phase, whereby the phase diagram depends on the strength of the Vicsek-like interactions. Employing a subsystem-block-density distribution analysis, we determine the critical point and make an attempt to estimate the critical exponents. In contrast to the passive model, we find that the critical point is not located on the rectilinear diameter. A first estimate of the critical exponents ß and ν is consistent with the underlying 3d-Ising universality class observed for the passive model.

Anomalous Fluctuations of Nematic Order in Solutions of Semiflexible Polymers.

The nematic ordering in semiflexible polymers with contour length L exceeding their persistence length ℓ_{p} is described by a confinement of the polymers in a cylinder of radius r_{eff} much larger than the radius r_{ρ} expected from the respective concentration of the solution. Large-scale molecular dynamics simulations combined with density functional theory are used to locate the isotropic-nematic (I-N) transition and to validate this cylindrical confinement. Anomalous fluctuations due to chain deflections from neighboring chains in the nematic phase are proposed. Considering deflections as collective excitations in the nematically ordered phase of semiflexible polymers elucidates the origins of shortcomings in the description of the I-N transition by existing theories.

A new insight into the isotropic-nematic phase transition in lyotropic solutions of semiflexible polymers: density-functional theory tested by molecular dynamics.

Semiflexible polymers in solution are studied for a wide range of both contour length L and persistence length lp as a function of monomer concentration under good solvent conditions. Both density-functional theory (DFT) and molecular dynamics (MD) simulation methods are used, and a very good agreement between both techniques is observed for rather stiff polymers. Evidence for a new mechanism of order parameter fluctuations in the nematic phase is presented, namely collective deformations of bundles of wormlike chains twisted around each other, and the typical wavelengths and amplitudes of these modes are estimated. These long wavelength fluctuations cause a reduction of the order parameter in comparison with the DFT prediction. It is also found that DFT becomes unreliable for rather flexible polymers in predicting that the transition from the isotropic (I)-phase to the nematic (N)-phase still exists at very high monomer concentrations (which in reality does not occur). However, under conditions when DFT is accurate, it provides reliable predictions also for the width of the I-N two-phase coexistence region, which are difficult to obtain from MD in spite of the use of very large systems (up to 500 000 monomers) by means of graphics processing units (GPU). For short and not very stiff chains, a pre-transitional chain stretching is found in the isotropic phase near the I-N-transition, not predicted by theories. A comparison with theoretical predictions by Khokhlov-Semenov, Odijk, and Chen reveals that the scaled transition densities are not simply functions of L/lp only, as these theories predict, but depend on d/lp (where d is the chain diameter) as well. Chain properties in the nematically ordered phase are compared to those of chains confined in tubes, and the deflection length concept is tested. Eventually, some consequences for the interpretation of experiments are spelled out.

Overview: Understanding nucleation phenomena from simulations of lattice gas models.

Monte Carlo simulations of homogeneous and heterogeneous nucleation in Ising/lattice gas models are reviewed with an emphasis on the general insight gained on the mechanisms by which metastable states decay. Attention is paid to the proper distinction of particles that belong to a cluster (droplet), that may trigger a nucleation event, from particles in its environment, a problem crucial near the critical point. Well below the critical point, the lattice structure causes an anisotropy of the interface tension, and hence nonspherical droplet shapes result, making the treatment nontrivial even within the conventional classical theory of homogeneous nucleation. For temperatures below the roughening transition temperature facetted crystals rather than spherical droplets result. The possibility to find nucleation barriers from a thermodynamic analysis avoiding a cluster identification on the particle level is discussed, as well as the question of curvature corrections to the interfacial tension. For the interpretation of heterogeneous nucleation at planar walls, knowledge of contact angles and line tensions is desirable, and methods to extract these quantities from simulations will be mentioned. Finally, also the problem of nucleation near the stability limit of metastable states and the significance of the spinodal curve will be discussed, in the light of simulations of Ising models with medium range interactions.

Dynamics of single semiflexible polymers in dilute solution.

We study the dynamics of a single semiflexible chain in solution using computer simulations, where we systematically investigate the effect of excluded volume, chain stiffness, and hydrodynamic interactions. We achieve excellent agreement with previous theoretical considerations, but find that the crossover from the time τb, up to which free ballistic motion of the monomers describes the chain dynamics, to the times W-1 or τ0, where anomalous monomer diffusion described by Rouse-type and Zimm-type models sets in, requires two decades of time. While in the limit of fully flexible chains the visibility of the anomalous diffusion behavior is thus rather restricted, the t3/4 power law predicted for stiff chains without hydrodynamic interactions is verified. Including hydrodynamics, evidence for the predicted [tln(t)]3/4 behavior is obtained. Similar good agreement with previous theoretical predictions is found for the decay of the bond autocorrelation functions and the end-to-end vector correlation. Finally, several predictions on the variation of characteristic relaxation times with persistence length describing the chain stiffness are tested.

On the polymer physics origins of protein folding thermodynamics.

A remarkable feature of the spontaneous folding of many small proteins is the striking similarity in the thermodynamics of the folding process. This process is characterized by simple two-state thermodynamics with large and compensating changes in entropy and enthalpy and a funnel-like free energy landscape with a free-energy barrier that varies linearly with temperature. One might attribute the commonality of this two-state folding behavior to features particular to these proteins (e.g., chain length, hydrophobic/hydrophilic balance, attributes of the native state) or one might suspect that this similarity in behavior has a more general polymer-physics origin. Here we show that this behavior is also typical for flexible homopolymer chains with sufficiently short range interactions. Two-state behavior arises from the presence of a low entropy ground (folded) state separated from a set of high entropy disordered (unfolded) states by a free energy barrier. This homopolymer model exhibits a funneled free energy landscape that reveals a complex underlying dynamics involving competition between folding and non-folding pathways. Despite the presence of multiple pathways, this simple physics model gives the robust result of two-state thermodynamics for both the cases of folding from a basin of expanded coil states and from a basin of compact globule states.