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.

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.

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.

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.

Blends of Semiflexible Polymers: Interplay of Nematic Order and Phase Separation.

Mixtures of semiflexible polymers with a mismatch in either their persistence lengths or their contour lengths are studied by Density Functional Theory and Molecular Dynamics simulation. Considering lyotropic solutions under good solvent conditions, the mole fraction and pressure is systematically varied for several cases of bending stiffness κ (the normalized persistence length) and chain length N. For binary mixtures with different chain length (i.e., NA=16, NB=32 or 64) but the same stiffness, isotropic-nematic phase coexistence is studied. For mixtures with the same chain length (N=32) and large stiffness disparity (κB/κA=4.9 to 8), both isotropic-nematic and nematic-nematic unmixing occur. It is found that the phase diagrams may exhibit a triple point or a nematic-nematic critical point, and that coexisting phases differ appreciably in their monomer densities. The properties of the two types of chains (nematic order parameters, chain radii, etc.) in the various phases are studied in detail, and predictions on the (anisotropic) critical behavior near the critical point of nematic-nematic unmixing are made.

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.

Phase Separation and Nematic Order in Lyotropic Solutions: Two Types of Polymers with Different Stiffnesses in a Common Solvent.

The interplay of the isotropic-nematic transition and phase separation in lyotropic solutions of two types of semiflexible macromolecules with pronounced difference in chain stiffness is studied by Density Functional Theory and Molecular Dynamics simulations. While the width of the isotropic-nematic two-phase coexistence region is narrow for solutions with a single type of semiflexible chain, the two-phase coexistence region widens for solutions containing two types of chains with rather disparate stiffness. In the nematic phase, both types of chains contribute to the nematic order, with intermediate values of the order parameter compared to the corresponding single component solutions. As the difference in bending stiffness is increased, the two chain types separate into two coexisting nematic phases. The phase behavior is rationalized by considering the chemical potentials of the two components and the Gibbs excess free energy. The geometric properties of the chain conformations under the various conditions are also discussed.

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.

Dynamic Responsive Formation of Nanostructured Fibers in a Hydrogel Network: A Molecular Dynamics Study.

In an effort to study natural fiber formation, such as, e.g., spider silk, we present a model, which is capable of forming biomimetic fibrillar nanostructure from a hydrogel micellar network. The latter consists of interacting atomic groups which form cores of micelles, and of flexible chains forming the shells of the micelles. Micelles are connected in a compact network by linearly stretched chains. The structural elements of the network can be transformed during deformation from micellar into fibrillary type and their evolution is found to depend significantly on strain rate. Our model suggests a set of conditions suitable for the formation of nanostructured fibrillar network. It demonstrates that a fibrillar structure is only formed upon sufficiently fast stretching while, in contrast, the micellar gel structure is preserved, if the material is pulled slowly. We illustrate this key aspect by a minimalistic model of only four chains as part of the whole network, which provides a detailed view on the mechanism of fibril formation. We conclude that such a simplified structure has similar functionality and is probably responsible for the formation of nano-structured molecular fibrils in natural materials.

Semiflexible Polymers Interacting With Planar Surfaces: Weak versus Strong Adsorption.

Semiflexible polymers bound to planar substrates by a short-range surface potential are studied by Molecular Dynamics simulations to clarify the extent to which these chain molecules can be considered as strictly two-dimensional. Applying a coarse-grained bead-spring model, the chain length N and stiffness κ as well as the strength of the adsorption potential ϵ w a l l are varied over a wide range. The excluded-volume (EV) interactions inherent in this model can also be "switched off" to provide a discretized version of the Kratky-Porod wormlike chain model. We study both local order parameters (fraction f of monomers within the range of the potential, bond-orientational order parameter η ) and the mean square gyration radius parallel, 〈 R g 2 〉 | | , and perpendicular, 〈 R g 2 〉 ⟂ , to the wall. While for strongly adsorbed chains EV has negligible effect on f and η , we find that 〈 R g 2 〉 | | is strongly affected when the chain contour length exceeds the persistence length. Monomer coordinates in perpendicular (⟂) direction are correlated over the scale of the deflection length which is estimated. It is found that f , η , and 〈 R g 2 〉 ⟂ converge to their asymptotic values with 1 / N corrections. For both weakly and strongly adsorbed chains, the distribution functions of "loops", "trains", and "tails" are analyzed. Some consequences pertaining to the analysis of experiments on adsorbed semiflexible polymers are pointed out.

Entropic Unmixing in Nematic Blends of Semiflexible Polymers.

Binary mixtures of semiflexible polymers with the same chain length, but different persistence lengths, separate into two coexisting different nematic phases when the osmotic pressure of the lyotropic solution is varied. Molecular Dynamics simulations and Density Functional Theory predict phase diagrams either with a triple point, where the isotropic phase coexists with two nematic phases or a critical point of unmixing within the nematic mixture. The difference in locally preferred bond angles between the constituents drives this unmixing without any attractive interactions between monomers.

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.

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.

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.

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.

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.

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.