Antidurotaxis Droplet Motion onto Gradient Brush Substrates.

Durotaxis motion is a spectacular phenomenon manifesting itself by the autonomous motion of a nano-object between parts of a substrate with different stiffness. This motion usually takes place along a stiffness gradient from softer to stiffer parts of the substrate. Here, we propose a new design of a polymer brush substrate that demonstrates antidurotaxis droplet motion, that is, droplet motion from stiffer to softer parts of the substrate. By carrying out extensive molecular dynamics simulation of a coarse-grained model, we find that antidurotaxis is solely controlled by the gradient in the grafting density of the brush and is favorable for fluids with a strong attraction to the substrate (low surface energy). The driving force of the antidurotaxial motion is the minimization of the droplet-substrate interfacial energy, which is attributed to the penetration of the droplet into the brush. Thus, we anticipate that the proposed substrate design offers a new understanding and possibilities in the area of autonomous motion of droplets for applications in microfluidics, energy conservation, and biology.

Unidirectional Droplet Propulsion onto Gradient Brushes without External Energy Supply.

Using extensive molecular dynamics simulation of a coarse-grained model, we demonstrate the possibility of sustained unidirectional motion (durotaxis) of droplets without external energy supply when placed on a polymer brush substrate with stiffness gradient in a certain direction. The governing key parameters for the specific substrate design studied, which determine the durotaxis efficiency, are found to be the grafting density of the brush and the droplet adhesion to the brush surface, whereas the strength of the stiffness gradient, the viscosity of the droplet, or the length of the polymer chains of the brush have only a minor effect on the process. It is shown that this durotaxial motion is driven by the steady increase of the interfacial energy between droplet and brush as the droplet moves from softer to stiffer parts of the substrate whereby the mean driving force gradually declines with decreasing roughness of the brush surface. We anticipate that our findings indicate further possibilities in the area of nanoscale motion without external energy supply.

Concave polymer brushes inwardly grafted in spherical cavities.

The structure and scaling properties of inwardly curved polymer brushes, tethered under good solvent conditions to the inner surface of spherical shells such as membranes and vesicles, are studied by extensive molecular dynamics simulations and compared with earlier scaling and self-consistent field theory predictions for different molecular weights of the polymer chains N and grafting densities σg in the case of strong surface curvature, R-1. We examine the variation of the critical radius R*(σg), separating the regimes of weak concave brushes and compressed brushes, predicted earlier by Manghi et al. [Eur. Phys. J. E 5, 519-530 (2001)], as well as various structural properties such as the radial monomer- and chain-end density profiles, orientation of bonds, and brush thickness. The impact of chain stiffness, κ, on concave brush conformations is briefly considered as well. Eventually, we present the radial profiles of the local pressure normal, PN, and tangential, PT, to the grafting surface, and the surface tension γ(σg), for soft and rigid brushes, and find a new scaling relationship PN(R)∝σg 4, independent of the degree of chain stiffness.

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.

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.

Nanoparticle diffusion in polymer melts: Molecular dynamics simulations and mode-coupling theory.

Nanoparticle diffusion in polymer melts is studied by the combination of Molecular Dynamics (MD) simulations and Mode-Coupling Theory (MCT). In accord with earlier experimental, simulation, and theoretical studies, we find that the Stokes-Einstein (SE) hydrodynamic relation Dn ∼ 1/Rn holds when the nanoparticle radius Rn is greater than the polymer gyration radius Rg, while in the opposite regime, the measured nanoparticle diffusion coefficient Dn exceeds the SE value by as much as an order of magnitude. The MCT values of Dn are found to be consistently higher than the MD simulation values. The observed discrepancy is attributed to the approximations involved in constructing the microscopic friction as well as to the approximate forms for dynamic structure factors used in MCT. In a thorough test of underlying MCT assumptions and approximations, various structural and dynamical quantities required as input for MCT are obtained directly from MD simulations. We present the improved MCT approach, which involves splitting of the microscopic time-dependent friction into two terms: binary (originating from short-time dynamics) and collective (due to long-time dynamics). Using MD data as input in MCT, we demonstrate that the total friction is largely dominated by its binary short-time term, which, if neglected, leads to severe overestimation of Dn. As a result, the revised version of MCT, in agreement with the present MD data, predicts 1/Rn 2 scaling of the probe diffusion coefficient in a non-hydrodynamic regime when Rn < Rg. If the total friction is dominated by the collective long-time component, one would observe 1/Rn 3 scaling of Dn in accordance with previous studies.

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}.

Anomalous Slowdown of Polymer Detachment Dynamics on Carbon Nanotubes.

The "wrapping" of polymer chains on the surface of carbon nanotubes allows one to obtain multifunctional hybrid materials with unique properties for a wide range of applications in biomedicine, electronics, nanocomposites, biosensors, and solar cell technologies. We study by means of molecular dynamics simulations the force-assisted desorption kinetics of a polymer from the surface of a carbon nanotube. We find that, due to the geometric coupling between the adsorbing surface and the conformation of the macromolecule, the process of desorption slows down dramatically upon increasing the windings around the nanotube. This behavior can be rationalized in terms of an overdamped dynamics with a frictional force that increases exponentially with the number of windings of the macromolecule, resembling the Euler-Eytelwein mechanism that describes the interplay between applied tension and frictional forces on a rope wrapped around a curved surface. The results highlight the fundamental role played by the geometry to control the dynamics and mechanical stability of hybrid materials in order to tailor properties and maximize performance.

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.

Stiffness-guided motion of a droplet on a solid substrate.

A range of technologies require the directed motion of nanoscale droplets on solid substrates. A way of realizing this effect is durotaxis, whereby a stiffness gradient of a substrate can induce directional motion without requiring an energy source. Here, we report on the results of extensive molecular dynamics investigations of droplets on a surface with varying stiffness. We find that durotaxis is enhanced by increasing the stiffness gradient and, also, by increased wettability of the substrate, in particular, when the droplet size decreases. We anticipate that our study will provide further insights into the mechanisms of nanoscale directional motion.

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.

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.

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.

Molecular weight effects on interfacial properties of linear and ring polymer melts: A molecular dynamics study.

Using molecular dynamics simulations, we study and compare the pressure, P, and the surface tension, γ, of linear chains and of ring polymers at the hard walls confining both melts into a slit. We examine the dependence of P and γ on the length (i.e., molecular weight) N of the macromolecules. For linear chains, we find that both pressure and surface tension are inversely proportional to the chain length, P(N)-P(N→∞)∝N-1,γ(N)-γ(N→∞)∝N-1, irrespective of whether the confining planes attract or repel the monomers. In contrast, for melts comprised of cyclic (ring) polymers, neither the pressure nor the surface tension is found to depend on molecular weight N for both kinds of wall-monomer interactions. While other structural properties as, e.g., the probability distributions of trains and loops at impenetrable walls appear quantitatively indistinguishable, we observe an amazing dissimilarity in the probability to find a chain end or a tagged monomer of a ring at a given distance from the wall in both kinds of polymeric melts. In particular, we demonstrate that the conformational equivalence of linear chains in a confined melt to a single chain under conditions of critical adsorption to a planar surface, established two decades ago, does also hold for ring polymers in a melt of linear chains. This analogy does not hold, however, for linear and ring chains in a confined melt of ring chains.

Semiflexible polymers under good solvent conditions interacting with repulsive walls.

Solutions of semiflexible polymers confined by repulsive planar walls are studied by density functional theory and molecular dynamics simulations, to clarify the competition between the chain alignment favored by the wall and the depletion caused by the monomer-wall repulsion. A coarse-grained bead-spring model with a bond bending potential is studied, varying both the contour length and the persistence length of the polymers, as well as the monomer concentration in the solution (good solvent conditions are assumed throughout, and solvent molecules are not included explicitly). The profiles of monomer density and pressure tensor components near the wall are studied, and the surface tension of the solution is obtained. While the surface tension slightly decreases with chain length for flexible polymers, it clearly increases with chain length for stiff polymers. Thus, at fixed density and fixed chain length, the surface tension also increases with increasing persistence length. Chain ends always are enriched near the wall, but this effect is much larger for stiff polymers than for flexible ones. Also the profiles of the mean square gyration radius components near the wall and the nematic order parameter are studied to clarify the conditions where wall-induced nematic order occurs.

Semiflexible polymer brushes and the brush-mushroom crossover.

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 = Na < lp, 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.