Probing chain ends in adsorbed polymer layers

We study theoretically the pull-out of polymer chains from an adsorbed polymer layer by sticking of the chain ends on an opposing surface using scaling arguments and a mean field theory. When only one chain is pulled out from the layer, we extend previous results obtained for a single adsorbed chain and calculate the force necessary to extract the chain from the layer. We then discuss end adsorption from an adsorbed layer of polymers bearing specific end groups onto a second surface. Two bridging regimes are predicted: a diffuse layer regime at weak separations (or/and weak interaction) and a large separation strong interaction regime where the bridges stretch into a brush like structure. Bridging fractions and force profiles are displayed that could be compared to atomic force microscope or surface force apparatus experiments.     

Stability of growing polymer chain ends for nitroxide-mediated photo-living radical polymerization

The stability of the growing polymer chain ends for the nitroxide-mediated photo-living radical polymerization of methyl methacrylate (MMA) was explored through block copolymerization with isopropyl methacrylate ( ⁱ PMA). The block copolymerization of ⁱ PMA was performed with the PMMA prepolymer prepared by the photopolymerization of MMA using the racemic-(2RS,2′RS)-azobis(4-methoxy-2,4-dimethylvaleronitrile) (r-AMDV) as the initiator, 4-methoxy-2,2,6,6-tetramethylpiperidine-1-oxyl (MTEMPO) as the mediator, and (4-tert-butylphenyl)-diphenylsulfonium triflate ( ^t BuS) as the photo-acid generator. When the polymerization of MMA was carried out for 6.5 h, the resulting block copolymer showed a bimodal GPC due to the deactivation of part of the growing chain ends of the prepolymer. On the other hand, when the MMA polymerization was shortened to 5 h, the unimodal block copolymer was obtained without deactivation of the prepolymer.     

Viscoelasticity of a stretched polymer chain with ends exposed to a constant force

The viscoelasticity of a stretched polymer chain with its end particles exposed to oppositely acting forces is studied via collisional molecular dynamics and analytically. A simple model according to which a polymer molecule is a chain of particles linked through freely jointed elastic bonds is adopted. The analytical theory is in good agreement with the results of the computer simulation of time correlation functions in the range of large-scale motions of a polymer molecule. It is found that the decay of correlation functions K _αβμν of fluctuations of the microscopic stress tensor of a chain, K _zzzz, K _zαzα , = K _zαzα , (α = x, y; z is the axis along which the forces act), is slowed down, and their value increases relative to the respective correlation functions of a chain with fixed ends. The greater the force, the higher this difference. The correlation functions that are transverse with respect to the z axis do not differ from those for chains with fixed ends. The results show that, in the calculation of time correlation functions of strongly stretched polymer chains, different statistical ensembles are not equivalent; this must be taken into account in the dynamic theories of heavily deformed polymers.     

Mean first passage times for bond formation for a Brownian particle in linear shear flow above a wall

Motivated by cell adhesion in hydrodynamic flow, here the authors study bond formation between a spherical Brownian particle in linear shear flow carrying receptors for ligands covering the boundary wall. They derive the appropriate Langevin equation which includes multiplicative noise due to position-dependent mobility functions resulting from the Stokes equation. They present a numerical scheme which allows to simulate it with high accuracy for all model parameters, including shear rate and three parameters describing receptor geometry (distance, size, and height of the receptor patches). In the case of homogeneous coating, the mean first passage time problem can be solved exactly. In the case of position-resolved receptor-ligand binding, they identify different scaling regimes and discuss their biological relevance.     

Dynamical integration of a Markovian web: a first passage time approach

In this work we address the dynamics of Markovian systems by tracking the evolution of the probability distribution, utilizing mean first passage time theory to augment the set of states considered. The method is validated on a lattice system and is applied, in conjunction with landscape analysis (saddle point searches) and multidimensional transition-state theory, to an atomistic model of glassy atactic polystyrene, in order to follow its time evolution over more than ten orders of magnitude on the time scale, from less than 10(-15) up to 10(-5) s. Frequencies extracted from the eigenvalues of the rate constant matrix are in favorable agreement with experimental measurements of subglass relaxation transitions at 250 K.     

An incremental mean first passage analysis for a quasistatic model of polymer translocation through a nanopore

For the translocation of a polymer through a nanopore, a quasistatic assumption for the dynamics yields a tractable form for the entropic barrier. Although this is a much simplified model, interesting features such as robust scaling emerge from its application. To explore these details, we present a method of mapping the translocation process as an incremental mean first passage problem. In this approach, the quantity of interest is the average first time t(0) at which the polymer achieves a displacement of Δs in the translocation coordinate s. Constructing scenarios with different initial conditions and boundary conditions, analytic and exact numerical approaches are used to resolve the dynamics of translocation in detail and generate new insight into the nature of the entropic barrier.     

Kinetic theory of binary nucleation based on a first passage time analysis

The binary classical nucleation theory (BCNT) is based on the Gibbsian thermodynamics and applies the macroscopic concept of surface tension to nanosize clusters. This leads to severe inconsistencies and large discrepancies between theoretical predictions and experimental results regarding the nucleation rate. We present an alternative approach to the kinetics of binary nucleation which avoids the use of classical thermodynamics for clusters. The new approach is an extension to binary mixtures of the kinetic theory previously developed by Narsimhan and Ruckenstein and Ruckenstein and Nowakowski [J. Colloid Interface Sci. 128, 549 (1989); 137, 583 (1990)] for unary nucleation which is based on molecular interactions and in which the rate of emission of molecules from a cluster is determined via a mean first passage time analysis. This time is calculated by solving the single-molecule master equation for the probability distribution of a "surface" molecule moving in a potential field created by the cluster. The starting master equation is a Fokker-Planck equation for the probability distribution of a surface molecule with respect to its phase coordinates. Owing to the hierarchy of characteristic time scales in the evolution of the molecule, this equation can be reduced to the Smoluchowski equation for the distribution function involving only the spatial coordinates. The new theory is combined with density functional theory methods to determine the density profiles. This is essential for nucleation in binary systems particularly when one of the components is surface active. Knowing these profiles, one can determine the potential fields created by the cluster, its rate of emission of molecules, and the nucleation rate more accurately than by using the uniform density approximation. The new theory is illustrated by numerical calculations for a model binary mixture of Lennard-Jones monomers and rigidly bonded dimers of Lennard-Jones atoms. The amphiphilic character of the dimer component (i.e., its surface activity) is induced by the asymmetry in the interaction between a monomer and the two different sites of a dimer. The inconsistencies of the BCNT are avoided in the new theory.     

Water diffusion through a membrane protein channel: a first passage time approach

Water diffusion through OmpF, a porin in the outer membrane of Escherichia coli, is studied by molecular dynamics simulation. A first passage time approach allows characterizing the diffusive properties of a well-defined region of this channel. A carbon nanotube, which is considerably more homogeneous, serves as a model to validate the methodology. Here we find, in addition to the expected regular behavior, a gradient of the diffusion coefficient at the channel ends, witness of the transition from confinement in the channel to bulk behavior in the connected reservoirs. Moreover, we observe the effect of a kinetic boundary layer, which is the counterpart of the initial ballistic regime in a mean square displacement analysis. The overall diffusive behavior of water in OmpF shows remarkable similarity with that in a homogeneous channel. However, a small fraction of the water molecules appears to be trapped by the protein wall for considerable lengths of time. The distribution of trapping times exhibits a broad power law distribution psi(tau) approximately tau (-2.4), up to tau=10 ns, a bound set by the length of the simulation run. We discuss the effect of this distribution on the dynamic properties of water in OmpF in terms of incomplete sampling of phase space.     

Distribution of chain ends at the surface of a polymer melt: Compensation effects and surface tension

We consider the surface of a nearly incompressible polymer melt, extending the usual ground-state analysis of self-consistent field theory to describe finite length polymers in the ground-state potential. To maintain self-consistency, further corrections to the potential are calculated within linear response theory. From this, we find an excess of ends near the surface, followed by a compensating depletion on the R_g length scale, which relies crucially on the finite compressibility of the melt. The attraction of ends to the surface can be described as resulting from a surface potential for ends with a strength on the order of k_BT. Our results address the long-standing controversies of the distribution of chain ends, the chain-length dependence of the surface tension, and the interaction between objects immersed in a melt. © 1995 John Wiley & Sons, Inc.     

Calculation of the mean first passage time tested on simple two-dimensional models

A particle diffusing in a two-dimensional (2D) container, shaped as a simplified configuration space of two passing 2D circular particles in a flat channel, is considered. The mean first passage time through one absorbing boundary is calculated using the one-dimensional Fick-Jacobs equation and its modification; both derived by mapping the 2D diffusion equation onto the longitudinal ("reaction") coordinate. The obtained results are compared with the hopping time, defined as the inverted lowest eigenvalue of the full 2D problem. The comparison shows that the mapped equations give reliable results, in contrast to predictions of the simplest concept of the transition state theory.     

Approximate first passage time distribution for barrier crossing in a double well under fractional Gaussian noise

The distribution of waiting times, f(t), between successive turnovers in the catalytic action of single molecules of the enzyme beta-galactosidase has recently been determined in closed form by Chaudhury and Cherayil [J. Chem. Phys. 125, 024904 (2006)] using a one-dimensional generalized Langevin equation (GLE) formalism in combination with Kramers' flux-over-population approach to barrier crossing dynamics. The present paper provides an alternative derivation of f(t) that eschews this approach, which is strictly applicable only under conditions of local equilibrium. In this alternative derivation, a double well potential is incorporated into the GLE, along with a colored noise term representing protein conformational fluctuations, and the resulting equation transformed approximately to a Smoluchowski-type equation. f(t) is identified with the first passage time distribution for a particle to reach the barrier top starting from an equilibrium distribution of initial points, and is determined from the solution of the above equation using local boundary conditions. The use of such boundary conditions is necessitated by the absence of definite information about the precise nature of the boundary conditions applicable to stochastic processes governed by non-Markovian dynamics. f(t) calculated in this way is found to have the same analytic structure as the distribution calculated by the flux-over-population method.     

Kinetic theory of nucleation based on a first passage time analysis: improvement by the density-functional theory

A recent kinetic theory of nucleation [see, e.g., E. Ruckenstein and B. Nowakowski, J. Colloid Interface Sci. 137, 583 (1990)] is based on molecular interactions and avoids the traditional thermodynamics. The rate of emission of molecules from a cluster is found via a first passage time analysis. This time is calculated by solving the single-molecule master equation for the probability distribution function of a surface molecule located in the potential field created by the cluster. The liquid cluster was assumed to have sharp boundaries and uniform density. In the present paper, this assumption is removed by using the density-functional theory to find the density profiles. Thus, more accurate calculations of the potential field created by the cluster, its emission rate, and nucleation rate are obtained. The modified theory is illustrated by numerical calculations for a molecular pair interaction potential combining the dispersive attraction with the hard-sphere repulsion.     

Bisphosphonate units in the main polymer chain: The first synthesis

Radical copolymerization of tetraethyl vinylidene phosphonate ( B ) with vinyl monomers has been described for the first time. In copolymerization with vinyl acetate ( V ) strictly alternating copolymer was formed even when [ V ]₀/[ B ]₀ was equal to 80. In copolymerization with acrylic acid ( A ) copolymers of the general structure ‐[( B )₁ A _x)]_n‐ were formed. The number of A units (x) was shown to depend on the [ A ]₀/[ B ]₀ ratio in the monomers feed. The reactivity ratio r_A was determined as equal to 2.1 and on this basis, the distribution of x as a function of [ A ]₀/[ B ]₀ was found. Bisphosphonic units were deblocked and the corresponding polyacids were analyzed by NMR spectra. M_n > 0.5 × 10⁶ were measured by SEC for copolymers of B with A . © 2012 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2012     

Mean field theory for a reversibly crosslinked polymer network

We present a mean field theory for melts and solutions of reversibly crosslinked polymers. In our model, crosslinks are considered as local bonds between two monomers. For a blend of A+B+AB polymers, we assume reversible crosslinks between the copolymers AB with a crosslink strength z and interaction weights ω(A) and ω(B) for monomers of type A and B, respectively. The usual mean field model for polymer blends without reversible crosslinks is recovered if z vanishes. With or without crosslinks, the A+B+AB blend can form a lamellar phase with A and B rich regions. If reversible crosslinks are enabled and ω(A) differs strongly from ω(B), the lamellar nanophase separation of A and B monomers is accompanied by a similar segregation of crosslinked and noncrosslinked polymers. If ω(A) and ω(B) are equal, crosslinked copolymers are well mixed with the homopolymers. For a homopolymer solution with reversible crosslinks between the polymers, our calculations show that polymers and solvent molecules are separated macroscopically if the Flory-Huggins interaction parameter and the crosslink strength are suitably high or if the volume fraction of polymers or the chain length are suitably low.     

Dynamic behavior of polymer surface and the time dependence of contact angle

The increased attention has been focused on the re-searches of soft materials proposed by Pierre-Gilles de Gennes, a Nobel Prize Laureate in Physics. A special issue of “Science” on soft surfaces was published in 2002 to review specific surface properti…     

Efficient computation of the first passage time distribution of the generalized master equation by steady-state relaxation

The generalized master equation or the equivalent continuous time random walk equations can be used to compute the macroscopic first passage time distribution (FPTD) of a complex stochastic system from short-term microscopic simulation data. The computation of the mean first passage time and additional low-order FPTD moments can be simplified by directly relating the FPTD moment generating function to the moments of the local FPTD matrix. This relationship can be physically interpreted in terms of steady-state relaxation, an extension of steady-state flow. Moreover, it is amenable to a statistical error analysis that can be used to significantly increase computational efficiency. The efficiency improvement can be extended to the FPTD itself by modelling it using a gamma distribution or rational function approximation to its Laplace transform.     

First passage time distribution of chaperone driven polymer translocation through a nanopore: homopolymer and heteropolymer cases

Combining the advection-diffusion equation approach with Monte Carlo simulations we study chaperone driven polymer translocation of a stiff polymer through a nanopore. We demonstrate that the probability density function of first passage times across the pore depends solely on the Pe?clet number, a dimensionless parameter comparing drift strength and diffusivity. Moreover it is shown that the characteristic exponent in the power-law dependence of the translocation time on the chain length, a function of the chaperone-polymer binding energy, the chaperone concentration, and the chain length, is also effectively determined by the Pe?clet number. We investigate the effect of the chaperone size on the translocation process. In particular, for large chaperone size, the translocation progress and the mean waiting time as function of the reaction coordinate exhibit pronounced sawtooth-shapes. The effects of a heterogeneous polymer sequence on the translocation dynamics is studied in terms of the translocation velocity, the probability distribution for the translocation progress, and the monomer waiting times.     

Using an incremental mean first passage approach to explore the viscosity dependent dynamics of the unbiased translocation of a polymer through a nanopore

Noting the limitations of the standard characterization of translocation dynamics, an incremental mean first passage process methodology is used to more completely map the unbiased translocation of a polymer through a nanopore. In this approach, the average time t(0) required to reach successively increasing displacements for the first time is recorded - a measure shown to be more commensurate with the mean first passage nature of translocation. Applying this methodology to the results of Langevin dynamics simulations performed in three dimensions across a range of viscosities, a rich set of dynamics spanning regular diffusion at low viscosities to sub-diffusion at higher viscosities is revealed. Further, while the scaling of the net translocation time τ with polymer length N is shown to be viscosity-dependent, common regimes are found across all viscosities: super-diffusive behaviour at short times, an N-independent backbone consistent with τ ～ N(2.0) at low viscosities and τ ～ N(2.2) at higher viscosities for intermediate times, and N-dependent deviations from the backbone near the completion of translocation.