Dynamical scaling of single chains on adsorbing substrates: diffusion processes

We study the dynamics of tethered chains of length N on adsorbing surfaces, considering the dilute case; for this we use the bond fluctuation model and scaling concepts. In particular, we focus on the mean-square displacement of single monomers and of the center of mass of the chains. The characteristic time tau of the fluctuations of a free chain in a good solvent grows as tau approximately N(a), where the coefficient a obeys a=2nu+1. We show that the same coefficient also holds at the critical point of adsorption. At intermediate time scales single monomers show subdiffusive behavior; this concurs with the behavior calculated from scaling arguments based on the dynamical exponent a. In the adsorbed state tau(perpendicular), the time scale for the relaxation in the direction perpendicular to the surface, becomes independent of N; tau(perpendicular) is then the relaxation time of an adsorption blob. In the direction parallel to the surface the motion is similar to that of a two-dimensional chain and is controlled by a time scale given by tau(parallel) approximately N(2nu(2)+1)L(-2Delta(nu/nu)), where nu(2) is the Flory exponent in two dimensions, nu is the Flory exponent in three dimensions, and Deltanu=nu(2)-nu. For the motion parallel to the surface we find dynamical scaling over a range of about four decades in time.     

Adsorption of semiflexible block copolymers on homogeneous surfaces

We present the results of extensive numerical off-lattice Monte Carlo simulations of semiflexible block-copolymer chains adsorbed onto flat homogeneous surfaces. We have compared the behavior of several chain structures, such as homopolymers, diblocks, (A(alpha)B(alpha)) block copolymers, and random heteropolymers. In all the cases studied, we have found the adsorption process to be favored with an increase of the chain rigidity. Particularly, the adsorption of diblock structures becomes a two-step process characterized by two different adsorbing temperatures that depend on the chain stiffness kappa, the chain length N, and the adsorbing energies epsilon(A) and epsilon(B). This twofold adsorbing process changes to a single one for copolymers of reduced block size alpha. Each block of the stiff copolymer chain is found to satisfy the classical scaling laws for flexible chains, however, we found the scaling exponent phi to depend on the chain stiffness. The measurement of the radius of gyration exhibits a typical behavior of a polymer chain composed of Nl(p) blobs whose persistence length follows l(p) approximately (kappa/k(B)T)(0.5) for large stiff chains.     

Polymer chains in a soft nanotube: a Monte Carlo study

We study the equilibrium properties of flexible polymer chains confined in a soft tube by means of extensive Monte Carlo simulations. The tube wall is that of a single sheet six-coordinated self-avoiding tethered membrane. Our study assumes that there is no adsorption of the chain on the wall. By varying the length N of the polymer and the tube diameter D we examine the variation of the polymer gyration radius Rg and diffusion coefficient Ddiff in soft and rigid tubes of identical diameter and compare them to scaling theory predictions. We find that the swollen region of the soft tube surrounding the chain exhibits a cigarlike cylindrical shape for sufficiently narrow tubes with D相似文献   

The critical adsorption point of self-avoiding walks: a finite-size scaling approach

The critical adsorption of self-avoiding polymer chain in a simple cubic lattice onto a flat surface is studied with Monte Carlo simulations. The dependence of number of surface contacts M on chain length N and polymer-surface interaction epsilon is investigated by a finite-size scaling approach. We estimate the critical adsorption point epsilon(c)=0.291+/-0.002 and the exponent phi=0.54+/-0.01. The asymptotic behaviors M proportional variant N for epsilon>epsilon(c) and M proportional variant N(0) for epsilon相似文献   

Confinement free energy and chain conformations of homopolymers confined between two repulsive walls

Lattice Monte Carlo simulations of polymer solutions confined between two parallel plates were performed. The confinement free energy Deltamicro(conf) per chain and the radius of gyrations of the chains parallel and perpendicular to the plates were obtained. When the concentration of the confined solution is above the overlap concentration, Deltamicro(conf) is found to scale with Na/D in a power law, betaDeltamicro(conf) approximately (Na/D)(m), with an exponent m=1.10+/-0.02 for athermal walls where N is the number of monomers in a chain, D is the slit width, and a is the lattice spacing. The presence of a weak attractive polymer/wall interaction epsilon(w) does not change the scaling variable, but the exponent m increases slightly. Extrapolating the results to melt would suggest that the predictions made by de Gennes [C. R. Acad. Sci. Paris II 305, 1181 (1987)] about the confinement free energy cost per chain in polymer melt is correct as far as the scaling variable is concerned, but is incorrect about the exponent m observed. The implication of this result on the predicted force between plates immersed in polymer melt is discussed. The parallel dimensions of the confined chain is expanded when the slit width D is narrow, however, the expansion is reduced at high concentration. It is conceivable that in melt the chain is not expanded when confined in a repulsive slit.     

Polymer translocation through a nanopore induced by adsorption: Monte Carlo simulation of a coarse-grained model

Dynamic Monte Carlo simulation of a bead-spring model of flexible macromolecules threading through a very narrow pore in a very thin rigid membrane are presented, assuming at the cis side of the membrane a purely repulsive monomer-wall interaction, while the trans side is attractive. Two choices of monomer-wall attraction epsilon are considered, one choice is slightly below and the other slightly above the "mushroom to pancake" adsorption threshold epsilon(c) for an infinitely long chain. Studying chain lengths N=32, 64, 128, and 256 and varying the number of monomers N(trans) (time t=0) that have already passed the pore when the simulation started, over a wide range, we find for epsilonepsilon(c) a finite number N(trans)(t=0) suffices that the translocation probability is close to unity. In the case epsilonepsilon(c), we find that the translocation time scales as tau proportional, variant N(1.65+/-0.08). We suggest a tentative scaling explanation for this result. Also the distribution of translocation times is obtained and discussed.     

Concentration and saturation effects of tethered polymer chains on adsorbing surfaces

We consider end-grafted chains at an adsorbing surface under good solvent conditions using Monte Carlo simulations and scaling arguments. Grafting of chains allows us to fix the surface concentration and to study a wide range of surface concentrations from the undersaturated state of the surface up to the brushlike regime. The average extension of single chains in the direction parallel and perpendicular to the surface is analyzed using scaling arguments for the two-dimensional semidilute surface state according to Bouchaud and Daoud [J. Phys. (Paris) 48, 1991 (1987)]. We find good agreement with the scaling predictions for the scaling in the direction parallel to the surface and for surface concentrations much below the saturation concentration (dense packing of adsorption blobs). Increasing the grafting density we study the saturation effects and the oversaturation of the adsorption layer. In order to account for the effect of excluded volume on the adsorption free energy we introduce a new scaling variable related with the saturation concentration of the adsorption layer (saturation scaling). We show that the decrease of the single chain order parameter (the fraction of adsorbed monomers on the surface) with increasing concentration, being constant in the ideal semidilute surface state, is properly described by saturation scaling only. Furthermore, the simulation results for the chains' extension from higher surface concentrations up to the oversaturated state support the new scaling approach. The oversaturated state can be understood using a geometrical model which assumes a brushlike layer on top of a saturated adsorption layer. We provide evidence that adsorbed polymer layers are very sensitive to saturation effects, which start to influence the semidilute surface scaling even much below the saturation threshold.     

Polymer translocation through a nanopore under an applied external field

We investigate the dynamics of polymer translocation through a nanopore under an externally applied field using the two-dimensional fluctuating bond model with single-segment Monte Carlo moves. We concentrate on the influence of the field strength E, length of the chain N, and length of the pore L on forced translocation. As our main result, we find a crossover scaling for the translocation time tau with the chain length from tau approximately N2nu for relatively short polymers to tau approximately N1+nu for longer chains, where nu is the Flory exponent. We demonstrate that this crossover is due to the change in the dependence of the translocation velocity v on the chain length. For relatively short chains v approximately N-nu, which crosses over to v approximately N(-1) for long polymers. The reason for this is that with increasing N there is a high density of segments near the exit of the pore, which slows down the translocation process due to slow relaxation of the chain. For the case of a long nanopore for which R parallel, the radius of gyration Rg along the pore, is smaller than the pore length, we find no clear scaling of the translocation time with the chain length. For large N, however, the asymptotic scaling tau approximately N1+nu is recovered. In this regime, tau is almost independent of L. We have previously found that for a polymer, which is initially placed in the middle of the pore, there is a minimum in the escape time for R parallel approximately L. We show here that this minimum persists for weak fields E such that EL is less than some critical value, but vanishes for large values of EL.     

A quantitative study of tethered chains in various solution conditions using Langmuir diblock copolymer monolayers

This article summarizes our investigations of tethered chain systems using Langmuir monolayers of poly(dimethylsiloxane)‐polystyrene (PDMS‐PS) diblock copolymers on organic liquids. In this system, the PDMS block adsorbs strongly to the air surface while the PS block dangles into the subphase liquid. The air surface can be made either repulsive or attractive for the tethered PS chain segments by choosing a subphase liquid which has a surface tension less than or greater than that of PS, respectively. The segment profile of the PS block is determined by neutron reflection as a function of the surface density, the molecular weights of the PS and PDMS blocks, and the solution conditions. We cover the range of reduced surface density (Σ ) characteristic of the large body of data in the literature for systems of chains tethered onto solid surfaces from dilute solution in good or theta solvent conditions (Σ < 12). We emphasize quantitative comparisons with analytical profile forms and scaling predictions. We find that the strong‐stretching limit assumed in analytical self‐consistent field calculations (SCF) and scaling theories is not valid over this Σ range. On the other hand, over a large portion of this range (Σ ⪇ 5) tethered chain profiles are well described by a renormalization group theory for weakly interacting or noninteracting chains. Simultaneous with the study of the profile form, the free energy of the tethered chains is examined through the surface tension. A strong increase in the surface pressure is observed with increasing surface density which determines the maximum surface density which can be achieved. This effect is attributed to a combination of higher order osmotic interactions and configurational constraints. This effect may explain several outstanding discrepancies regarding the adsorption of end‐functionalized chains and diblock copolymers onto solid surfaces.     

Polymer brushes in solvents of variable quality: molecular dynamics simulations using explicit solvent

The structure and thermodynamic properties of a system of end-grafted flexible polymer chains grafted to a flat substrate and exposed to a solvent of variable quality are studied by molecular dynamics methods. The macromolecules are described by a coarse-grained bead-spring model, and the solvent molecules by pointlike particles, assuming Lennard-Jones-type interactions between pairs of monomers (epsilon(pp)), solvent molecules (epsilon(ss)), and solvent monomer (epsilon(ps)), respectively. Varying the grafting density sigma(g) and some of these energy parameters, we obtain density profiles of solvent particles and monomers, study structural properties of the chain (gyration radius components, bond orientational parameters, etc.), and examine also the profile of the lateral pressure P( parallel)(z), keeping in the simulation the normal pressure P( perpendicular) constant. From these data, the reduction of the surface tension between solvent and wall as a function of the grafting density of the brush has been obtained. Further results include the stretching force on the monomer adjacent to the grafting site and its variation with solvent quality and grafting density, and dynamic characteristics such as mobility profiles and chain relaxation times. Possible phase transitions (vertical phase separation of the solvent versus lateral segregation of the polymers into "clusters," etc.) are discussed, and a comparison to previous work using implicit solvent models is made. The variation of the brush height and the interfacial width of the transition zone between the pure solvent and the brush agrees qualitatively very well with corresponding experiments.     

Conformational studies of bottle-brush polymers absorbed on a flat solid surface

The adsorption of a bottle-brush polymer end-grafted with one chain end of its backbone to a flat substrate surface is studied by Monte Carlo simulation of a coarse-grained model, that previously has been characterized in the bulk, assuming a dilute solution under good solvent conditions. Applying the bond fluctuation model on the simple cubic lattice, we vary the backbone chain length N(b) from N(b)=67 to N(b)=259 effective monomeric units, the side chain length N from N=6 to N=48, and set the grafting density to σ=1, i.e., parameters that correspond well to the experimentally accessible range. When the adsorption energy strength ? is varied, we find that the adsorption transition (which becomes well-defined in the limit N(b)→∞, for arbitrary finite N) roughly occurs at the same value ?(c) as for ordinary linear chains (N=0), at least within our statistical errors. Mean square end-to-end distances and gyration radii of the side chains are obtained, as well as the monomer density profile in the direction perpendicular to the adsorbing surface. We show that for longer side chains the adsorption of bottle-brushes is a two-step process, the decrease of the perpendicular linear dimension of side chains with adsorption energy strength can even be nonmonotonic. Also, the behavior of the static structure factor S(q) is analyzed, evidence for a quasi-two-dimensional scaling is presented, and consequences for the interpretation of experiments are discussed.     

Some remarks on the dynamics of polymer melts

Considering one long chain (N monomeric units) in a homodisperse melt of chemically identical, but shorter, “solvent” chains (P monomers per chain), we propose some tentative scaling laws for the self-diffusion constant D(N) and the relaxation time T(N) of the solute chain. We also discuss the viscosity increment δ_η due to a small volume fraction Φ of the long chains. We find three regimes of behavior, depending on N and P, and on the distance between entanglement points (assumed smaller than N and P): (A) reptation of the N chain; (B) Stokes–Einstein regime; the solute moves like a usual polymer coil in a viscous fluid of P chains; (C) mixed regime, where D(N) is controlled by reptation, while δ_η is of type B. Contrary to our earlier belief, we find no significant regime where the process of “tube renewal” could be dominant.     

Computer simulation of bottle-brush polymers with flexible backbone: good solvent versus theta solvent conditions

By molecular dynamics simulation of a coarse-grained bead-spring-type model for a cylindrical molecular brush with a backbone chain of N(b) effective monomers to which with grafting density σ side chains with N effective monomers are tethered, several characteristic length scales are studied for variable solvent quality. Side chain lengths are in the range 5 ≤ N ≤ 40, backbone chain lengths are in the range 50 ≤ N(b) ≤ 200, and we perform a comparison to results for the bond fluctuation model on the simple cubic lattice (for which much longer chains are accessible, N(b) ≤ 1027, and which corresponds to an athermal, very good, solvent). We obtain linear dimensions of the side chains and the backbone chain and discuss their N-dependence in terms of power laws and the associated effective exponents. We show that even at the theta point the side chains are considerably stretched, their linear dimension depending on the solvent quality only weakly. Effective persistence lengths are extracted both from the orientational correlations and from the backbone end-to-end distance; it is shown that different measures of the persistence length (which would all agree for Gaussian chains) are not mutually consistent with each other and depend distinctly both on N(b) and the solvent quality. A brief discussion of pertinent experiments is given.     

Polymer translocation through a nanopore: a two-dimensional Monte Carlo study

We investigate the problem of polymer translocation through a nanopore in the absence of an external driving force. To this end, we use the two-dimensional fluctuating bond model with single-segment Monte Carlo moves. To overcome the entropic barrier without artificial restrictions, we consider a polymer which is initially placed in the middle of the pore and study the escape time tau required for the polymer to completely exit the pore on either end. We find numerically that tau scales with the chain length N as tau approximately N(1+2nu), where nu is the Flory exponent. This is the same scaling as predicted for the translocation time of a polymer which passes through the nanopore in one direction only. We examine the interplay between the pore length L and the radius of gyration R(g). For LR(g), we find tau approximately N. In addition, we numerically find the scaling function describing crossover between short and long pores. We also show that tau has a minimum as a function of L for longer chains when the radius of gyration along the pore direction R( parallel) approximately L. Finally, we demonstrate that the stiffness of the polymer does not change the scaling behavior of translocation dynamics for single-segment dynamics.     

An exact enumeration study on adsorption transition of a tethered chain at an adsorbing surface

<正>The number of configurations,c(n,m),of a single chain with length n attached to a flat surface with m monomers contacting the surface is exactly enumerated.A function of c(n,m) about m and n is obtained.From the function,a scaling law for mean energy of chain is derived,and we estimate the critical pointε_c = 0.276 and the crossover exponentφ= 0.5.The free energy difference between tethered chain and free chain in dilute solution is also studied,which shows the critical adsorption point is about 0.272 for infinite long chain with φ= 0.5.     

The colloidal force of bead-spring chains in a good solvent

A recently developed density functional theory (DFT) for tethered bead-spring chains is used to investigate colloidal forces for the good solvent case. A planar surface of tethered chains is opposed to a bare, hard wall and the force exerted on the bare wall is calculated by way of the contact density. Previously, the case of large wall separation was investigated. The density profiles of the unperturbed chains, in that case, were found to be neither stepfunctions nor parabolas and were shown to accurately predict computer simulation results. In the present paper, the surface forces that result from the distortion of these density profiles at finite wall separation is studied. The resulting force function is analyzed for varying surface coverages, wall separations, and chain lengths. The results are found to be in near quantitative agreement with the scaling predictions of Alexander [S. Alexander, J. Phys. (Paris) 38, 983 (1977)] when the layer thickness is "correctly" defined. Finally, a hybrid Alexander-DFT theory is suggested for the analysis of experimental results.     

Conformational statistics of polymer chain terminally attached to wall (II)

The SAW tail chains were studied. The permitted conformational number and the mean square end-to-end distance as a function of the chain length N for such a model tail chain were obtained by computer simulations, including the exact enumeration and Monte Carlo method. These two basic quantities obeyed the relations deduced from the scaling law. The critical exponents and the lattice indexes were given by fitting the data of the computer experiments. It has been shown that there is a certain extension in the size of the SAW tail chains as well as the NRW tail chains in the direction normal to the wall. The normal component of the mean square end-to-end distance is almost twice as large as the parallel component of the short chain SAW. However, as N →∞, the effect of the wall on the chain conformation becomes a little weak because of the self-avoiding behavior for the model. That is quite different from the case of the NRW tail chain. Project supported by the National Natural Science Foundation of China     

Unmixing of polymer blends confined in ultrathin films: crossover between two-dimensional and three-dimensional behavior

The interplay between chain conformations and phase separation in binary symmetric polymer mixtures confined into thin films by "neutral" hard walls (i.e., walls that do not preferentially attract or repel one of the two components of the mixture) is studied by Monte Carlo simulations. Using the bond fluctuation model on a simple cubic lattice in the semi grand canonical ensemble, we locate the critical temperature of demixing via finite size scaling methods for a wide range of chain lengths (16 infinity, and hence T(c) proportional, variant N. However, strong deviations from the Flory-Huggins theory occur as long as the unperturbed chain dimension exceeds D, and the critical behavior falls in the universality class of the two-dimensional Ising model for any finite value of D.     

Polysoaps in Backbone‐Selective Solvents: Effects of Side‐Chain Length on Collapse Dynamics

Abstract

Molecular dynamics (MD) simulations of model comb‐graft heteropolymers were performed to understand general mechanistic features of coil‐to‐micelle relaxation after instantaneous quench from a nonselective solvent to solvent conditions selective for the backbone monomers and poor for the side‐chain monomers. The systems considered were single bead‐spring molecules with backbones of 30 monomers and 10 equally spaced side chains of 1, 5, 10, or 20 monomers each, immersed in dense liquids of 20,000 simple solvent particles. We find that the coil‐to‐micelle relaxation time, τ _r , averaged over 50 independent trajectories for each set of topological parameters considered, decreases with increasing side‐chain length. A two‐stage relaxation mechanism is observed: (1) a fast collapse and aggregation of neighboring side chains to form a chain of “protomicelles,” followed by (2) a slow intramolecular aggregation of protomicelles. Fast collapse dominates for molecules with relatively longer side chains due to relatively higher probabilities of initial contacts between side‐chain monomers in different side chains, while slow intramolecular aggregation dominates for molecules with relatively shorter side chains.     

Irreversible adsorption of tethered chains at substrates: Monte Carlo study

The irreversible adsorption of single chains grafted with one end to the surface is studied using scaling arguments and computer simulations. We introduce a two-phase model, in which the chain is described by an adsorbate portion and a corona portion formed by nonadsorbed monomers. The adsorption process can be viewed as consisting of a main stage, during which monomers join by "zipping" (along their order in the chain) the surface, and a late stage, in which the remaining corona collapses on the surface. Based on our model we derive a scaling relation for the time of adsorption t(M) as a function of the number M of adsorbed monomers; t(M) follows a power law, M(alpha), with alpha > 1. We find that alpha is related to the Flory exponent nu by alpha = 1 + nu. Using further scaling arguments we derive relations between the overall time of adsorption, the characteristic time of adsorption (given by the crossover time between the main and the last stage of adsorption), and the chain length. To support our analysis we perform Monte Carlo simulations using the bond fluctuation model. In particular, the sequence of adsorption events is very well reproduced by the simulations, and an analysis of the various density profiles supports our theoretical model. Especially the loop formation during adsorption clearly shows that the growth of the adsorbate is dominated by zipping. The simulations are also in almost quantitative agreement with our theoretical scaling analysis, showing that here the assumption of a linear relation between Monte Carlo steps and time is well obeyed. We conclude by also discussing the geometrical shape of the adsorbate.