Modification of Continuum Chain Model of Surface-Interacting Polymers To Describe the Crossover between Weak and Strong Adsorption (original) (raw)
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Reconciling lattice and continuum models for polymers at interfaces
The Journal of Chemical Physics, 2012
It is well known that lattice and continuum descriptions for polymers at interfaces are, in principle, equivalent. In order to compare the two models quantitatively, one needs a relation between the inverse extrapolation length c as used in continuum theories and the lattice adsorption parameter χ s (defined with respect to the critical point). So far, this has been done only for ideal chains with zero segment volume in extremely dilute solutions. The relation χ s (c) is obtained by matching the boundary conditions in the two models. For depletion (positive c and χ s ) the result is very simple: χ s = ln(1 + c/5). For adsorption (negative c and χ s ) the ideal-chain treatment leads to an unrealistic divergence for strong adsorption: c decreases without bounds and the train volume fraction exceeds unity. This due to the fact that for ideal chains the volume filling cannot be accounted for. We extend the treatment to real chains with finite segment volume at finite concentrations, for both good and theta solvents. For depletion the volume filling is not important and the ideal-chain result χ s = ln(1 + c/5) is generally valid also for non-ideal chains, at any concentration, chain length, or solvency. Depletion profiles can be accurately described in terms of two length scales: ρ = tanh 2 [(z + p)/δ], where the depletion thickness (distal length) δ is a known function of chain length and polymer concentration, and the proximal length p is a known function of c (or χ s ) and δ. For strong repulsion p = 1/c (then the proximal length equals the extrapolation length), for weaker repulsion p depends also on chain length and polymer concentration (then p is smaller than 1/c). In very dilute solutions we find quantitative agreement with previous analytical results for ideal chains, for any chain length, down to oligomers. In more concentrated solutions there is excellent agreement with numerical self-consistent depletion profiles, for both weak and strong repulsion, for any chain length, and for any solvency. For adsorption the volume filling dominates. As a result c now reaches a lower limit c ≈ −0.5 (depending slightly on solvency). This limit follows immediately from the condition of a fully occupied train layer. Comparison with numerical SCF calculations corroborates that our analytical result is a good approximation. We suggest some simple methods to determine the interaction parameter (either c or χ s ) from experiments. The relation χ s (c) provides a quantitative connection between continuum and lattice theories, and enables the use of analytical continuum results to describe the adsorption (and stretching) of lattice chains of any chain length. For example, a fully analytical treatment of mechanical desorption of a polymer chain (including the temperature dependence and the phase transitions) is now feasible.
The Journal of Chemical Physics, 2001
An ideal polymer chain anchored to a planar surface is considered by using both lattice and continuum model approaches. A general equation relating the lattice and continuum model adsorption interaction parameters is derived in a consistent way by substituting the exact continuum solution for the free chain end distribution function into the lattice model boundary condition. This equation is not mathematically exact but provides excellent results. With the use of this relation the quantitative equivalence between lattice and continuum results was demonstrated for chains of both infinite and finite length and for all three regimes corresponding to attractive, repulsive and adsorption-threshold energy of polymer-surface interaction. The obtained equations are used to discuss the distribution functions describing the tail of an anchored macromolecule and its adsorbed parts. For the tail-related properties the results are independent of the microscopic details of the polymer chain and the adsorbing surface. One interesting result obtained in the vicinity of adsorption threshold point is a bimodal tail length distribution function, which manifests chain populations with either tail or loop dominance. The properties related to the number of surface contacts contain, apart from universal scaling terms, also a nonuniversal factor depending on microscopic details of polymer-surface interaction. We derived an equation for calculating this nonuniversal factor for different lattice models and demonstrated excellent agreement between the lattice results and the continuum model.
Polymer Chain Adsorption on a Solid Surface: Scaling Arguments and Computer Simulations
Springer Series in Surface Sciences, 2010
We examine the phase transition of polymer adsorption as well as the underlying kinetics of polymer binding from dilute solutions on a structureless solid surface. The emphasis is put on the properties of regular multiblock copolymers, characterized by block size M and total length N as well as on random copolymers with quenched composition p of sticky and neutral segments. The macromolecules are modeled as coarse-grained bead-spring chains subject to a short-ranged surface adhesive potential. Phase diagrams, showing the variation of the critical threshold for single chain adsorption in terms of M and p are derived from scaling considerations in agreement with results from computer experiment.
The Journal of Chemical Physics, 1996
The adsorption of a single polymer chain onto a solid surface is investigated by molecular dynamics simulations. The chain is composed of mass points interacting via a truncated Lennard-Jones potential, i.e., the excluded volume interaction is taken into account, and grafted to the surface with one end. The average adsorption degree is calculated for various chain lengths ͑N ϭ 16, 32, 64, 128͒ and adsorption energies. In addition, the scaling behavior of the adsorption degree and the radius of gyration is investigated. The adsorption degree and the average length of loops and tails are obtained for chains of various stiffnesses. In this context, we find that stiffer chains adsorb more easily. Moreover, the distribution of the mass points perpendicular to the surface as well as the orientation of the bonds with respect to the surface is discussed for various adsorption energies and stiffnesses.
Chain-Length Dependence of the Polymer Surface Excess near the Adsorption/Depletion Transition
Macromolecules, 2002
The behavior of a solution of ideal polymers at a surface near the adsorption/depletion transition is investigated using a lattice model. A very simple relation was found for the surface excess θ ex of a polymer molecule of length N close to the adsorption/depletion transition: θ ex /φ b) A[B + (ssc ∞)N], where φ b is the bulk volume fraction of polymer, A ≈ 5/6 and B ≈ 1/5 are constants, s is the Silberberg adsorption parameter, and sc ∞ is the critical adsorption energy for infinite chain lengths. The adsorption/depletion transition shifts to lower s values with decreasing chain length. This effect is strongly enhanced if the end segments of the chain adsorb preferentially. A continuum model was used to obtain analytical expressions. The agreement between lattice and continuum descriptions is quite good. The chain-length dependence of the adsorption/depletion transition for homopolymers is not found in the continuum model, however. The results in this paper may be relevant for example for critical chromatography.
A dynamical Monte Carlo model of polymer adsorption
Macromolecules, 1993
A novel dynamical Monte Carlo computer model designed to investigate the configurational relaxation of homopolymers at the solid/solution interface is described. Ensembles of polymer chains, each up to 99 segments in length, have been generated by n-step random self-avoiding walks throughout a simple cubic lattice, the basal face of which represented an impenetrable solid surface. To model adsorption, each chain is generated "in solution" at the center of the lattice and then displaced to the surface and allowed to approach its equilibrium adsorbed conformation through a succession of elementary moves operating on randomly selected segments. Each chain was considered to be in an athermal solvent environment, and the effect of surface coverage was simulated by the inclusion of a periodic boundary constraint. When the enthalpy of adsorption is low, it has been possible to observe chains that have initially adsorbed, relaxed, completely desorbed, and then readsorbed. For isolated nonadsorbed chains, root-mean-square end-bend distance data from the model have provided an estimate of the universal scaling exponent, Y, of 0.691, which is in good agreement with the Flory value of 0.6 for polymers in athermal solvents. By introduction of a probability of desorption, the model hae also been used to study the evolution of the structure of the adsorbed layer for polymers undergoing both chemical (irreversibIe) and physical adsorption at the interface.
Surface adsorption and collapse transition of a linear polymer chain in three dimensions
Journal of Physics A: Mathematical and General, 1999
The critical behaviour of surface adsorption and collapse transition of a flexible selfattracting self-avoiding polymer chain is examined. Depending upon the underlying lattice and space dimensionality, phase diagrams that exhibit many different universality domains of critical behavior are found. We discuss these phase diagrams and the values of the critical exponents found from different theoretical methods.
Polymer adsorption on heterogeneous surfaces
The European Physical Journal B, 1998
The adsorption of a single ideal polymer chain on energetically heterogeneous and rough surfaces is investigated using a variational procedure introduced by Garel and Orland (Phys. Rev. B 55 (1997), 226). The mean polymer size is calculated perpendicular and parallel to the surface and is compared to the Gaussian conformation and to the results for polymers at flat and energetically homogeneous surfaces. The disorder-induced enhancement of adsorption is confirmed and is shown to be much more significant for a heterogeneous interaction strength than for spatial roughness. This difference also applies to the localization transition, where the polymer size becomes independent of the chain length. The localization criterion can be quantified, depending on an effective interaction strength and the length of the polymer chain.
Adsorption of a single polymer chain on a surface: Effects of the potential range
Physical Review E, 2013
We investigate the effects of the range of adsorption potential on the equilibrium behavior of a single polymer chain end-attached to a solid surface. The exact analytical theory for ideal lattice chains interacting with a planar surface via a box potential of depth U and width W is presented and compared to continuum model results and to Monte Carlo (MC) simulations using the pruned-enriched Rosenbluth method for self-avoiding chains on a simple cubic lattice. We show that the critical value U c corresponding to the adsorption transition scales as W −1/ν , where the exponent ν = 1/2 for ideal chains and ν ≈ 3/5 for self-avoiding walks. Lattice corrections for finite W are incorporated in the analytical prediction of the ideal chain theory U c ≈ ( π 2 24 )(W + 1/2) −2 and in the best-fit equation for the MC simulation data U c = 0.585(W + 1/2) −5/3 . Tail, loop, and train distributions at the critical point are evaluated by MC simulations for 1 W 10 and compared to analytical results for ideal chains and with scaling theory predictions. The behavior of a self-avoiding chain is remarkably close to that of an ideal chain in several aspects. We demonstrate that the bound fraction θ and the related properties of finite ideal and self-avoiding chains can be presented in a universal reduced form: θ(N,U,W ) = θ(NU c ,U/U c ). By utilizing precise estimations of the critical points we investigate the chain length dependence of the ratio of the normal and lateral components of the gyration radius. Contrary to common expectations this ratio attains a limiting universal value R 2 g⊥ / R 2 g = 0.320 ± 0.003 only at N ∼ 5000. Finite-N corrections for this ratio turn out to be of the opposite sign for W = 1 and for W 2. We also study the N dependence of the apparent crossover exponent φ eff (N ). Strong corrections to scaling of order N −0.5 are observed, and the extrapolated value φ = 0.483 ± 0.003 is found for all values of W . The strong correction to scaling effects found here explain why for smaller values of N , as used in most previous work, misleadingly large values of φ eff (N ) were identified as the asymptotic value for the crossover exponent.
Exact results for the adsorption of a semiflexible copolymer chain in three dimensions
Arxiv preprint arXiv:0909.4653, 2009
Lattice model of directed self avoiding walk has been solved analytically to investigate adsorption desorption phase transition behaviour of a semiflexible sequential copolymer chain on a two dimensional impenetrable surface perpendicular to the preferred direction of the walk of the copolymer chain in three dimensions. The stiffness of the chain has been accounted by introducing an energy barrier for each bend in the walk of the copolymer chain. Exact value of adsorption desorption transition points have been determined using generating function method for the cases in which one type of monomer is having interaction with the surface viz., (i) no interaction (ii) attractive interaction and (iii) repulsive interaction. Results obtained in each of the case show that for stiffer copolymer chain adsorption transition occurs at a smaller value of monomer surface attraction than a flexible copolymer chain. These features are similar to that of a semi-flexible homopolymer chain adsorption.