Polymer-mediated entropic forces between scale-free objects

Exploring the Molecular Origin for the Long-Range Propagation of the Substrate Effect in Unentangled Poly(methyl methacrylate) Films

The polymer/substrate interface plays a significant role in the dynamics of nanoconfined polymers because of its suppression on polymer mobility and its long-range propagation feature, while the molecular origin of the long-range substrate effect in unentangled polymer material is still ambiguous. Herein, we investigated the propagation distances of the substrate effect (h*) by a fluorinated tracer-labeled method of two unentangled polymer films supported on silicon substrates: linear and ring poly(methyl methacrylate) films with relatively low molecular weights. The results indicate that the value of h* has a molecular weight dependence of h*∝N (N is the degree of polymerization) in the unentangled polymer films, while h*∝N1/2 was presented as previously reported in the entangled films. A theoretical model, depending on the polymer/polymer intermolecular interaction, was proposed to describe the above long-range propagation behavior of the substrate effect and agrees with our experiment results very well. From the model, it revealed that the intermolecular friction determines the long-range propagation of the substrate effect in the unentangled system, but the intermolecular entanglement is the dominant role in entangled system. These results give us a deeper understanding of the long-range substrate effect.

Entropic force of cone-tethered polymers interacting with a planar surface

Computer simulations are used to characterize the entropic force of one or more polymers tethered to the tip of a hard conical object that interact with a nearby hard flat surface. Pruned-enriched Rosenbluth method Monte Carlo simulations are used to calculate the variation of the conformational free energy F of a hard-sphere polymer with respect to the cone-tip-to-surface distance h from which the variation of the entropic force f≡|dF/dh| with h is determined. We consider the following cases: a single freely jointed tethered chain, a single semiflexible tethered chain, and several freely jointed chains of equal length each tethered to the cone tip. The simulation results are used to test the validity of a prediction by Maghrebi et al. [Maghrebi et al., Europhys. Lett. 96, 66002 (2011)0295-507510.1209/0295-5075/96/66002; Phys. Rev. E 86, 061801 (2012)1539-375510.1103/PhysRevE.86.061801] that f∝(γ_{∞}-γ_{0})h^{-1}, where γ_{0} and γ_{∞} are universal scaling exponents for the partition function of the tethered polymer for h=0 and h=∞, respectively. The measured functions f(h) are generally consistent with the predictions, with small quantitative discrepancies arising from the approximations employed in the theory. In the case of multiple tethered polymers, the entropic force per polymer is roughly constant, which is qualitatively inconsistent with the predictions.

Free-energy cost of localizing a single monomer of a confined polymer

We describe a simple Monte Carlo simulation method to calculate the free-energy cost of localizing a single monomer of a polymer confined to a cavity. The localization position is chosen to be on the inside surface of the confining cavity. The method is applied to a freely jointed hard-sphere polymer chain confined to cavities of spherical and cubic geometries. In the latter case, we consider localization at a corner and at the center of a face of the confining cube. We consider cases of end-monomer localization both with and without tethering of the other end monomer to a point on the surface. We also examine localization of monomers at arbitrary positions along the contour of the polymer. We characterize the dependence of the free energy on the cavity size and shape, the localization position, and the polymer length. The quantitative trends can be understood using standard scaling arguments and use of a simple theoretical model. The results are relevant to those theories of polymer translocation that focus on the importance of the free-energy barrier as the translocation process requires an initial localization of a monomer to the position of a nanopore.

Measurement of entropic force from polymers attached to a pyramidal tip

The measurement of the boundary shape dependence of the entropic force from long polymers was attempted. The pyramidal cone-plate geometry was chosen. The polymer molecules were covalently bound to a well-defined Au patch at the apex of a pyramidal cantilever tip of the atomic force microscope (AFM). A smooth hydrophobic plate was used as the second boundary to confine the polymer molecules. The use of the hydrophobic plate allows neglect of polymer adhesion forces. The measurements were made in salt water solution to decrease the effect of electrostatic forces from any uncompensated charges on the boundary. As the functionalized AFM tip approaches the flat hydrophobic surface, the induced entropic forces were measured as a function of the separation distance. The measured force-distance curves are compared with a model of polymer-mediated entropic force between scale-free objects and the Alexander-de Gennes (AdG) theory for a polymer brush.

Localization of random walks to competing manifolds of distinct dimensions

We consider localization of a random walk (RW) when attracted or repelled by multiple extended manifolds of different dimensionalities. In particular, we consider a RW near a rectangular wedge in two dimensions, where the (zero-dimensional) corner and the (one-dimensional) wall have competing localization properties. This model applies also (as cross section) to an ideal polymer attracted to the surface or edge of a rectangular wedge in three dimensions. More generally, we consider (d-1)- and (d-2)-dimensional manifolds in d-dimensional space, where attractive interactions are (fully or marginally) relevant. The RW can then be in one of four phases where it is localized to neither, one, or both manifolds. The four phases merge at a special multicritical point where (away from the manifolds) the RW spreads diffusively. Extensive numerical analyses on two-dimensional RWs confined inside or outside a rectangular wedge confirm general features expected from a continuum theory, but also exhibit unexpected attributes, such as a reentrant localization to the corner while repelled by it.

Pinning and unbinding of ideal polymers from a wedge corner

A polymer repelled by unfavorable interactions with a uniform flat surface may still be pinned to attractive edges and corners. This is demonstrated by considering adsorption of a two-dimensional ideal polymer to an attractive corner of a repulsive wedge. The well-known mapping between the statistical mechanics of an ideal polymer and the quantum problem of a particle in a potential is then used to analyze the singular behavior of the unbinding transition of the polymer. The divergence of the localization length is found to be governed by an exponent that varies continuously with the angle (when reflex). Numerical treatment of the discrete (lattice) version of such an adsorption problem confirms this behavior.

Attractive and repulsive polymer-mediated forces between scale-free surfaces

We consider forces acting on objects immersed in, or attached to, long fluctuating polymers. The confinement of the polymer by the obstacles results in polymer-mediated forces that can be repulsive (due to loss of entropy) or attractive (if some or all surfaces are covered by adsorbing layers). The strength and sign of the force in general depends on the detailed shape and adsorption properties of the obstacles but assumes simple universal forms if characteristic length scales associated with the objects are large. This occurs for scale-free shapes (such as a flat plate, straight wire, or cone) when the polymer is repelled by the obstacles or is marginally attracted to it (close to the depinning transition where the absorption length is infinite). In such cases, the separation h between obstacles is the only relevant macroscopic length scale, and the polymer-mediated force equals Ak_{B}T/h, where T is temperature. The amplitude A is akin to a critical exponent, depending only on geometry and universality of the polymer system. The value of A, which we compute for simple geometries and ideal polymers, can be positive or negative. Remarkably, we find A=0 for ideal polymers at the adsorption transition point, irrespective of shapes of the obstacles, i.e., at this special point there is no polymer-mediated force between obstacles (scale free or not).

Nonequilibrium interactions between ideal polymers and a repulsive surface

We use Newtonian and overdamped Langevin dynamics to study long flexible polymers dragged by an external force at a constant velocity v. The work W performed by that force depends on the initial state of the polymer and the details of the process. The Jarzynski equality can be used to relate the nonequilibrium work distribution P(W) obtained from repeated experiments to the equilibrium free energy difference ΔF between the initial and final states. We use the power law dependence of the geometrical and dynamical characteristics of the polymer on the number of monomers N to suggest the existence of a critical velocity v_{c}(N), such that for v<v_{c} the reconstruction of ΔF is an easy task, while for v significantly exceeding v_{c} it becomes practically impossible. We demonstrate the existence of such v_{c} analytically for an ideal polymer in free space and numerically for a polymer which is being dragged away from a repulsive wall. Our results suggest that the distribution of the dissipated work W_{d}=W-ΔF in properly scaled variables approaches a limiting shape for large N.

Casimir interaction of rodlike particles in a two-dimensional critical system

We consider the fluctuation-induced interaction of two thin, rodlike particles, or "needles," immersed in a two-dimensional critical fluid of Ising symmetry right at the critical point. Conformally mapping the plane containing the needles onto a simpler geometry in which the stress tensor is known, we analyze the force and torque between needles of arbitrary length, separation, and orientation. For infinite and semi-infinite needles we utilize the mapping of the plane bounded by the needles onto the half plane, and for two needles of finite length we use the mapping onto an annulus. For semi-infinite and infinite needles the force is expressed in terms of elementary functions, and we also obtain analytical results for the force and torque between needles of finite length with separation much greater than their length. Evaluating formulas in our approach numerically for several needle geometries and surface universality classes, we study the full crossover from small to large values of the separation to length ratio. In these two limits the numerical results agree with results for infinitely long needles and with predictions of the small-particle operator expansion, respectively.

Long polymers near wedges and cones

We perform a Monte Carlo study of N-step self-avoiding walks, attached to the corner of an impenetrable wedge in two dimensions (d=2), or the tip of an impenetrable cone in d=3, of sizes ranging up to N=10(6) steps. We find that the critical exponent γ(α), which determines the dependence of the number of available conformations on N for a cone or wedge with opening angle α, is in good agreement with the theory for d=2. We study the end-point distribution of the walks in the allowed space and find similarities to the known behavior of random walks (ideal polymers) in the same geometry. For example, the ratio between the mean square end-to-end distances of a polymer near the cone or wedge and a polymer in free space depends linearly on γ(α), as is known for ideal polymers. We show that the end-point distribution of polymers attached to a wedge does not separate into a product of angular and radial functions, as it does for ideal polymers in the same geometry. The angular dependence of the end position of polymers near the wedge differs from theoretical predictions.

Analysis of Single Locus Trajectories for Extracting In Vivo Chromatin Tethering Interactions

Is it possible to extract tethering forces applied on chromatin from the statistics of a single locus trajectories imaged in vivo? Chromatin fragments interact with many partners such as the nuclear membrane, other chromosomes or nuclear bodies, but the resulting forces cannot be directly measured in vivo. However, they impact chromatin dynamics and should be reflected in particular in the motion of a single locus. We present here a method based on polymer models and statistics of single trajectories to extract the force characteristics and in particular when they are generated by the gradient of a quadratic potential well. Using numerical simulations of a Rouse polymer and live cell imaging of the MAT-locus located on the yeast Saccharomyces cerevisiae chromosome III, we recover the amplitude and the distance between the observed and the interacting monomer. To conclude, the confined trajectories we observed in vivo reflect local interaction on chromatin.

Diffusion in the presence of scale-free absorbing boundaries

Scale-free surfaces, such as cones, remain unchanged under a simultaneous expansion of all coordinates by the same factor. Probability density of a particle diffusing near such absorbing surface at large time approaches a simple form that incorporates power-law dependencies on time and distance from a special point, such as apex of the cone, which are characterized by a single exponent η. The same exponent is used to describe the number of spatial conformations of long ideal polymer attached to the special point of a repulsive surface of the same geometry and can be used in calculation of entropic forces between such polymers and surfaces. We use the solution of diffusion equation near such surfaces to find the numerical values of η, as well as to provide some insight into the behavior of ideal polymers near such surfaces.

Ideal polymers near scale-free surfaces

The number of allowed configurations of a polymer is reduced by the presence of a repulsive surface resulting in an entropic force between them. We develop a method to calculate the entropic force, and detailed pressure distribution, for long ideal polymers near a scale-free repulsive surface. For infinite polymers the monomer density is related to the electrostatic potential near a conducting surface of a charge placed at the point where the polymer end is held. Pressure of the polymer on the surface is then related to the charge density distribution in the electrostatic problem. We derive explicit expressions for pressure distributions and monomer densities for ideal polymers near a two- or three-dimensional wedge, and for a circular cone in three dimensions. Pressure of the polymer diverges near sharp corners in a manner resembling (but not identical to) the electric field divergence near conducting surfaces. We provide formalism for calculation of all components of the total force in situations without axial symmetry.

Critical Casimir torques and forces acting on needles in two spatial dimensions

We investigate the universal orientation-dependent interactions between nonspherical colloidal particles immersed in a critical solvent by studying the instructive paradigm of a needle embedded in bounded two-dimensional Ising models at bulk criticality. For a needle in an Ising strip, the interaction on mesoscopic scales depends on the width of the strip and the length, position, and orientation of the needle. By lattice Monte Carlo simulations we evaluate the free-energy difference between needle configurations being parallel and perpendicular to the strip. We concentrate on small but nonetheless mesoscopic needle lengths for which analytic predictions are available for comparison. All combinations of boundary conditions for the needles and boundaries are considered which belong to either the "normal" or the "ordinary" surface universality class, i.e., which induce local order or disorder, respectively. We also derive exact results for needles of arbitrary mesoscopic length, in particular for needles embedded in a half plane and oriented perpendicularly to the corresponding boundary as well as for needles embedded at the center line of a symmetric strip with parallel orientation.