We present exact spatio-temporal correlation functions of a Rouse polymer chain submerged in a fluid having planar mixed flow, in the steady state. Using these correlators, determination of the time scale distribution functions associated with the first-passage tumbling events is difficult in general; it was done recently in Phys. Rev. Lett.,101, 188301 (2008), for the special case of "simple shear" flow. We show here that the method used in latter paper fails for the general mixed flow problem. We also give many new estimates of the exponent $θ$ associated with the exponential tail of the angular tumbling time distribution in the case of simple shear
We study analytically a model of a two dimensional, partially directed, flexible or semiflexible polymer, attached to an attractive wall which is perpendicular to the preferred direction. In addition, the polymer is stretched by an externally applied force. We find that the wall has a dramatic effect on the polymer. For wall attraction smaller than the non-sequential nearest neighbor attraction, the fraction of monomers at the wall is zero and the model is the same as that of a polymer without a wall. However, for greater than, the fraction of monomers at the wall undergoes a first order trans…
We have mapped the physics of a system of semi-flexible inextensible polymers onto a complex Ginzburg-Landau field theory using techniques of functional integration. It is shown in the limit of low number density of monomers in a melt of semi-flexible, inextensible polymers, kept apart by an excluded volume interaction, the radius of gyration scales as N^{nu}, where N is the chain length, and nu = 1, in contrast to the value of nu ~ 0.6 for flexible polymer melts. Using Renormalization Group arguments, we show that the system exhibits an infra-red stable fixed point which can be identified as …
We consider random walk model of a semi-flexible polymer chain on a square and a cubic lattice to enumerate conformations of the polymer chain in two and three dimensions, respectively. The bending energy of the chain is assumed as the key factor which controls the minimum average length of the chain in between two successive bends in the chain; and the average length of the chains per unit bend is defined as the persistence length of the polymer chain. It has been found that the minimum energy required to bend the chain is expressed in the form of simple relation which includes space dimensio…
We consider the escape of a flexible, self-avoiding polymer chain out of a confined geometry. By means of simulations, we demonstrate that the translocation time can be described by a simple scaling law that exhibits a nonlinear dependence on the degree of polymerization and that is sensitive to the nature of the confining geometry. These results contradict earlier predictions but are in agreement with recently confirmed geometry-dependent expressions for the free energy of confinement.
Using a combination of the replica-exchange Monte Carlo algorithm and the multicanonical method, we investigate the influence of bending stiffness on the conformational phases of a bead-stick homopolymer model and present the pseudo-phase diagram for the complete range of semi-flexible polymers, from flexible to stiff. Although a simple model, we observe a rich variety of conformational phases, reminiscent of conformations observed for synthetic polymers or biopolymers. Changing the internal bending stiffness, the model exhibits different pseudo phases like bent, hairpin or toroidal. In partic…
We investigate the binding transition of two flexible polymers grafted to a steric surface with closeby end points. While free polymers show a discontinuous transition, grafting to a steric flat surface leads to a continuous binding transition. This is supported by results from Metropolis and parallel multicanonical simulations. A combination of canonical and microcanonical analyses reveals that the change in transition order can be understood in terms of the reduced translational entropy of the unbound high-temperature phase upon grafting.
We present exact and analytically accurate results for the problem of a flexible polymer chain in shear flow. Under such a flow the polymer tumbles, and the probability distribution of the tumbling times $τ$ of the polymer decays exponentially as $\sim \exp(-ατ/τ_0)$ (where $τ_0$ is the longest relaxation time). We show that for a Rouse chain, this nontrivial constant $α$ can be calculated in the limit of large Weissenberg number (high shear rate) and is in excellent agreement with our simulation result of $α\simeq 0.324$. We also derive exactly the distribution functions for the length and th…
We study the behavior of a flexible polymer chain in the presence of a low-molecular weight solvent in the vicinity of a liquid-gas critical point within the framework of a self-consistent field theory. The total free energy of the dilute polymer solution is expressed as a function of the radius of gyration of the polymer and the average solvent number density within the gyration volume at the level of the mean-field approximation. Varying the strength of attraction between polymer and solvent we show that two qualitatively different regimes occur at the liquid-gas critical point. In case of w…
We report Monte Carlo simulations of the self-assembly of supramolecular polymers based on a model of patchy particles. We find a first-order phase transition, characterized by hysteresis and nucleation, toward a solid bundle of polymers, of length much greater than the average gas phase length. We argue that the bundling transition is the supramolecular equivalent of the sublimation transition, that results from a weak chain-chain interaction. We provide a qualitative equation of state that gives physical insight beyond the specific values of the parameters used in our simulations.
Using a lattice-based Monte Carlo code for simulating self-avoiding flexible polymers in three dimensions in the absence of explicit hydrodynamics, we study their Rouse modes. For self-avoiding polymers, the Rouse modes are not expected to be statistically independent; nevertheless, we demonstrate that numerically these modes maintain a high degree of statistical independence. Based on high-precision simulation data we put forward an approximate analytical expression for the mode amplitude correlation functions for long polymers. From this, we derive analytically and confirm numerically severa…
We study the statistical mechanics of double-stranded semi-flexible polymers using both analytical techniques and simulation. We find a transition at some finite temperature, from a type of short range order to a fundamentally different sort of short range order. In the high temperature regime, the 2-point correlation functions of the object are identical to worm-like chains, while in the low temperature regime they are different due to a twist structure. In the low temperature phase, the polymers develop a kink-rod structure which could clarify some recent puzzling experiments on actin.
The quasi--equilibrium or maximum entropy approximation is applied in order to derive constitutive equations from kinetic models of polymer dynamics. It is shown in general and illustrated for an example how canonical distribution functions are obtained from the maximum entropy principle, how macroscopic and constitutive equations are derived therefrom and how these constitutive equations can be implemented numerically. In addition, a measure for the accuracy of the quasi--equilibrium approximation is proposed that can be evaluated while integrating the constitutive equations. In the example c…
We consider the lattice model for an ideal-linear polymer chain to mimic the conformations of the semi-flexible homo-polymer chain. The polymer chain is assumed to confine in the fairly small area, such that the flexible chain conformations are easily polymerized in the nano-area. It has been described using analytical calculations that such a situation may lead to interesting and distinct theoretical findings of the thermodynamics of semi-flexible homo-polymeric macromolecules where the macromolecules are confined in the nano-length scale. We also discuss the characteristics of the persistent…
Recently it has been shown that a two-dimensional model of self-attracting polymers based on attracting segments with the addition of stiffness displays three phases: a swollen phase, a globular, liquid-like phase, and an anisotropic crystal-like phase. Here, we consider the attracting segment model in three dimensions with the addition of stiffness. While we again identify a swollen and two distinct collapsed phases, we find that both collapsed phases are anisotropic, so that there is no phase in which the polymer resembles a disordered liquid drop. Moreover all the phase transitions are firs…
The development of a coherent conceptual basis for the treatment of non-linear microscopic phenomena, such as, hydrodynamic interaction, finite extensibility, excluded volume and internal viscosity, in molecular theories of dilute polymer solutions, is discussed. In particular, recent advances in the treatment of hydrodynamic interaction are reviewed, and the successive refinements which have ultimately led to the prediction of universal viscometric functions in theta solvents are highlighted.
We study the model of a partially directed flexible or semi-flexible homopolymer on a square lattice, subject to an externally applied force, in a direction either parallel to, or perpendicular to the preferred direction. The polymer is self-interacting and can therefore undergo a collapse transition. We show that this model can be solved and we obtain the force-temperature phase diagrams which, for the case of flexible polymers, agree with that of Brak et al obtained using a different method. At sufficiently low temperatures, the polymer conformation changes from compact to coil state as the …
We investigate the addition of stiffness to the lattice model of hydrogen-bonded polymers in two and three dimensions. We find that, in contrast to polymers that interact via a homogeneous short-range interaction, the collapse transition is unchanged by any amount of stiffness: this supports the physical argument that hydrogen bonding already introduces an effective stiffness. Contrary to possible physical arguments, favouring bends in the polymer does not return the model's behaviour to that comparable to the semi-flexible homogeneous interaction model, where the canonical $θ$-point occurs fo…
Polymer translocation in three dimensions out of planar confinements is studied in this paper. Three membranes are located at $z=-h$, $z=0$ and $z=h_1$. These membranes are impenetrable, except for the middle one at $z=0$, which has a narrow pore. A polymer with length $N$ is initially sandwiched between the membranes placed at $z=-h$ and $z=0$ and translocates through this pore. We consider strong confinement (small $h$), where the polymer is essentially reduced to a two-dimensional polymer, with a radius of gyration scaling as $R^{\tinytext{(2D)}}_g \sim N^{ν_{\tinytext{2D}}}$; here, $ν_{\ti…
We study the adsorption of semiflexible polymers such as polyelectrolytes or DNA on planar and curved substrates, e.g., spheres or washboard substrates via short-range potentials using extensive Monte-Carlo simulations, scaling arguments, and analytical transfer matrix techniques. We show that the adsorption threshold of stiff or semiflexible polymers on a planar substrate can be controlled by polymer stiffness: adsorption requires the highest potential strength if the persistence length of the polymer matches the range of the adsorption potential. On curved substrates, i.e., an adsorbing sphe…
We employ the recently introduced generalized microcanonical inflection point method for the statistical analysis of phase transitions in flexible and semiflexible polymers and study the impact of the bending stiffness upon the character and order of transitions between random-coil, globules, and pseudocrystalline conformations. The high-accuracy estimates of the microcanonical entropy and its derivatives required for this study were obtained by extensive replica-exchange Monte Carlo simulations. We observe that the transition behavior into the compact phases changes qualitatively with increas…
We model a melt of monodisperse side-chain liquid crystalline polymers as a melt of comb copolymers in which the side groups are rod-coil diblock copolymers. We consider both excluded volume and Maier-Saupe interactions. The first acts among any pair of segments while the latter acts only between rods. Using a free energy functional calculated from this microscopic model, we study the spinodal stability of the isotropic phase against density and orientational fluctuations. The phase diagram obtained in this way predicts nematic and smectic instabilities as well as the existence of microphases …
Using results from Schramm Loewner evolution (SLE), we give the expression of the fluctuation-induced force exerted by a polymer on a small impenetrable disk, in various 2-dimensional domain geometries. We generalize to two polymers and examine whether the fluctuation force can trap the object into a stable equilibrium. We compute the force exerted on objects at the domain boundary, and the force mediated by the polymer between such objects. The results can straightforwardly be extended to any SLE interface, including Ising, percolation, and loop-erased random walks. Some are relevant for extr…
We revisit the problem of semi-flexible chain condensation by neutral depleting agents (e.g. colloidal spheres or flexible polymers) by using a simple formalism that allows us to address its main features without specifying the kind of depleting agents. Correlations between depleting agents are shown to produce a reswelling of the chain at high enough volume fraction, consistent with an earlier analysis by Grosberg et al. (Biopolymers, 21:2413, 1982) in the context of DNA condensation induced by neutral polymers. It is also shown that the conditions under which spherical colloids can condense …
We present a theoretical description of the dynamics of a semi-flexible polymer being pulled through a nanopore by an external force acting at the pore. Our theory is based on the tensile blob picture of Pincus in which the front of the tensile force propagates through the backbone of the polymer, as suggested by Sakaue and recently applied to study a completely flexible polymer with self-avoidance, by Dubbledam et al. For a semi-flexible polymer with a persistence length P , its statistics is self-avoiding for a very long chain. As the local force increases, the blob size starts to decrease. …
Many theoretical works have attempted to coarse grain gene expression at the level of transcription and translation via frameworks based on exclusion processes. Usually in these models the three-dimensional conformation of the substrates (DNA and mRNA) is neglected, and particles move on a static unidimensional lattice in contact to an infinite reservoir. In this work we generalise the paradigmatic exclusion process and study the transport of particles along a unidimensional polymer-like flexible lattice immersed in a three-dimensional particle reservoir. We study the recycling of particles in…
For a self-repelling polymer chain consisting of n segments we calculate the persistence length L(j,n), defined as the projection of the end-to-end vector on the direction of the j`th segment. This quantity shows some pronounced variation along the chain. Using the renormalization group and epsilon-expansion we establish the scaling form and calculate the scaling function to order epsilon^2. Asymptotically the simple result L(j,n) ~ const(j(n-j)/n)^(2nu-1) emerges for dimension d=3. Also outside the excluded volume limit L(j,n) is found to behave very similar to the swelling factor of a chain …
The non-equilibrium structural and dynamical properties of semiflexible polymers confined to two dimensions are investigated by molecular dynamics simulations. Three different scenarios are considered: The force-extension relation of tethered polymers, the relaxation of an initially stretched semiflexible polymer, and semiflexible polymers under shear flow. We find quantitative agreement with theoretical predictions for the force-extension relation and the time dependence of the entropically contracting polymer. The semiflexible polymers under shear flow exhibit significant conformational chan…
Semi-flexible manifolds such as fluid membranes or semi-flexible polymers undergo delocalization transitions if they are subject to attractive interactions. We study manifolds with short-ranged interactions by field-theoretic methods based on the operator product expansion of local interaction fields. We apply this approach to manifolds in a random potential. Randomness is always relevant for fluid membranes, while for semi-flexible polymers there is a first order transition to the strong coupling regime at a finite temperature.
The surface tension of interacting polymers in a good solvent is calculated theoretically and by computer simulations for a planar wall geometry and for the insertion of a single colloidal hard-sphere. This is achieved for the planar wall and for the larger spheres by an adsorption method, and for smaller spheres by a direct insertion technique. Results for the dilute and semi-dilute regimes are compared to results for ideal polymers, the Asakura-Oosawa penetrable-sphere model, and to integral equations, scaling and renormalization group theories. The largest relative changes with density are …
We study the stochastic resonance (SR) of a flexible polymer chain crossing over a piecewise linear bistable potential. The dependence of signal to noise ratio $SNR$ on noise intensity $D$, coupling constant $k$ and polymer length $N$ is studied via two state approximation. We find that the response of signal to the background noise strength is significant at optimum values of $D_{opt}$, $k_{opt}$ and $N_{opt}$ which suggests novel means of manipulating proteins or vesicles. Furthermore, the thermally activated barrier crossing rate $r_{k}$ for the flexible polymer chain is studied. We find th…
A phase diagram for a surface-interacting long flexible polymer chain in a two-dimensional poor solvent where the possibility of collapse exists is determined using exact enumeration method. A model of a self-attracting self avoiding walk (SASAW) on a square lattice was considered and up to 28 steps in series were evaluated. A new adsorbed state having the conformation of a surface attached globule is found. Four phases (i) desorbed expanded, (ii) desorbed collapsed, (iii) adsorbed expanded and (iv) surface attached globule are found to meet at a point on the adsorption line.
We analyze the structural behavior of a single polymer chain grafted to an attractive, flexible surface. Our model is composed of a coarse-grained bead-and-spring polymer and a tethered membrane. By means of extensive parallel tempering Monte Carlo simulations it is shown that the system exhibits a rich phase behavior ranging from highly ordered, compact to extended random coil structures and from desorbed to completely adsorbed or even partially embedded conformations. These findings are summarized in a pseudophase diagram indicating the predominant class of conformations as a function of the…
We present a Bethe approximation to study lattice models of linear polymers. The approach is variational in nature and based on the cluster variation method (CVM). We focus on a model with $(i)$ a nearest neighbor attractive energy $ε_v$ between pair of non--bonded monomers, $(ii)$ a bending energy $ε_h$ for each pair of successive chain segments which are not collinear. We determine the phase diagram of the system as a function of the reduced temperature $t=\frac{T}{ε_v}$ and of the parameter $x=\frac{ε_h}{ε_v}$. We find two different qualitative behaviors, on varying $t$. For small values of…
Configurations of a single semiflexible polymer is studied when it is pushed into a nanochannel in the case where the polymer persistence length $l_p$ is much longer than the channel diameter $D$, i.e. $l_p/D \gg 1$. Using numerical simulations, we show that the polymer undergoes a sequence of recurring structural transitions upon longitudinal compression, i.e. random deflection along the channel, helix going around the channel wall, double-fold random deflection, double-fold helix, etc. We find that the helix transition can be understood as buckling of deflection segments, and the initial hel…
We consider the van der Waals interaction between a pair of polymers with quenched heterogeneous sequences of local polarizabilities along their backbones, and study the effective pairwise interaction energy for both stiff polymers and flexible Gaussian coils. In particular, we focus on the cases where the pair of polarizability sequences are (i)~distinct and (ii)~identical. We find that the pairwise interaction energies of distinct and identical Gaussian coils are both isotropic and exhibit the same decay behavior for separations larger than the gyration radius, in contradistinction to the or…
Linear Viscoelasticity of Semidilute Unentangled Flexible Polymer Solutions
IITB-Monash Research Academy, Indian Institute of Technology Bombay（IITB-莫纳什研究学院，印度理工学院孟买分校）；Department of Chemical Engineering, Indian Institute of Technology Bombay（印度理工学院孟买分校化学工程系）；Department of Chemical and Biological Engineering, Monash University（莫纳什大学化学与生物工程系）
For the estimation of transition points of finite elastic, flexible polymers with chain lengths from $13$ to $309$ monomers, we compare systematically transition temperatures obtained by the Fisher partition function zeros approach with recent results from microcanonical inflection-point analysis. These methods rely on accurate numerical estimates of the density of states, which have been obtained by advanced multicanonical Monte Carlo sampling techniques. Both the Fisher zeros method and microcanonical inflection-point analysis yield very similar results and enable the unique identification o…
Using a general Green function formulation, we re-derive, both, (i) Spitzer and his followers results for the winding angle distribution of the planar Brownian motion, and (ii) Edwards-Prager-Frisch results on the statistical mechanics of a ring polymer entangled with a straight bar. In the statistical mechanics part, we consider both cases of quenched and annealed topology. Among new results, we compute exactly the (expectation value of) the surface area of the locus of points such that each of them has linking number $n$ with a given closed random walk trajectory ($=$ ring polymer). We also …
We have studied the compact phase conformations of semi-flexible polymer chains confined in two dimensional nonhomogeneous media, modelled by fractals that belong to the family of modified rectangular (MR) lattices. Members of the MR family are enumerated by an integer $p$ $(2\leq p<\infty)$, and fractal dimension of each member of the family is equal to 2. The polymer flexibility is described by the stiffness parameter $s$, while the polymer conformations are modelled by weighted Hamiltonian walks (HWs). Applying an exact method of recurrence equations we have found the asymptotic behavior of…
The osmotic pressure $P$ in equilibrium polymers (EP) in good solvent is investigated by means of a three dimensional off-lattice Monte Carlo simulation. Our results compare well with real space renormalisation group theory and the osmotic compressibility $K \propto ϕ\upd ϕ/\upd P$ from recent light scattering study of systems of long worm-like micelles. We confirm the scaling predictions for EP based on traditional physics of quenched monodisperse polymers in the dilute and semidilute limit. Specifically, we find $P\propto ϕ^{2.3}$ and, hence, $K \propto ϕ^{-0.3}$ in the semidilute regime ---…
We report on a computational study of the statics and dynamics of long flexible linear polymers that spontaneously knot and unknot. Specifically, the equilibrium self-entanglement properties, such as the knotting probability, knot length and position, are investigated with extensive Monte Carlo sampling of chains of up to 15,000 beads. Tens of such equilibrated chains of up to 4, 096 beads are next used as starting points for Langevin dynamics simulations. The complex interplay of chain dynamics and self-knotting is addressed by monitoring the time evolution of various metric and entanglement …
This paper presents a theoretical analysis of the closure kinetics of a polymer with hydrodynamic interactions. This analysis, which takes into account the non-Markovian dynamics of the end-to-end vector and relies on the preaveraging of the mobility tensor (Zimm dynamics), is shown to reproduce very accurately the results of numerical simulations of the complete non linear dynamics. It is found that Markovian treatments, based on a Wilemski-Fixman approximation, significantly overestimate cyclization times (up to a factor 2), showing the importance of memory effects in the dynamics. In additi…
Polymer chains undergoing a continuous adsorption-desorption transition are studied through extensive computer simulations. A three-dimensional self-avoiding walk lattice model of a polymer chain grafted onto a surface has been treated for different solvent conditions. We have used an advanced contact-density chain-growth algorithm, in which the density of contacts can be directly obtained. From this quantity, the order parameter and its fourth-order Binder cumulant are computed, as well as the corresponding critical exponents and the adsorption-desorption transition temperature. As the number…
The conformational and dynamical properties of active self-propelled filaments/polymers are investigated in the presence of hydrodynamic interactions by both, Brownian dynamics simulations and analytical theory. Numerically, a discrete linear chain composed of active Brownian particles is considered, analytically, a continuous linear semiflexible polymer with active velocities changing diffusively. The force-free nature of active monomers is accounted for - no Stokeslet fluid flow induced by active forces - and higher order hydrodynamic multipole moments are neglected. The nonequilibrium chara…
We study a model of a semiflexible long chain polymer confined to a two-dimensional slit of width $w$, and interacting with the walls of the slit. The interactions with the walls are controlled by Boltzmann weights $a$ and $b$, and the flexibility of the polymer is controlled by another Boltzmann weight $c$. This is a simple model of the steric stabilisation of colloidal dispersions by polymers in solution. We solve the model exactly and compute various quantities in $(a,b,c)$-space, including the free energy and the force exerted by the polymer on the walls of the slit. In some cases these qu…
