Mapping of continuum and lattice models for describing the adsorption of an ideal chain anchored to a planar surface

The depletion thickness in solutions of semi-flexible polymers near colloidal surfaces: analytical approximations

We derive a simple, yet accurate approximate mean-field expression for the depletion thickness δsf of a solution of dilute semi-flexible polymers next to a hard surface. In the case of a hard wall this equation has the simple form δsf = δ0[1 - tanh(psf/δ0)], where psf accounts for the degree of flexibility and δ0 is the depletion thickness in the case of fully flexible polymers. For fixed polymer coil size, increasing the chain stiffness leads to a decrease in the depletion thickness. The approach is also extended to include higher polymer concentrations in the semidilute regime. The analytical expressions are in quantitative agreement with numerical self-consistent field computations. A remarkable finding is that there is a maximum in the depletion thickness as a function of the chain stiffness in the semidilute concentration regime. This also means that depletion attractions between colloidal particles reach a maximum for a certain chain stiffness, which may have important implications for the phase stability of colloid-polymer mixtures. The derived equations could be useful for the description of interactions in- and phase stability of mixtures of colloids and semi-flexible polymers.

Self-Consistent Field Modeling of Pulling a Test-Chain away from or Pushing It into a Polymer Adsorption Layer

We consider single chain force measurements to unravel characteristics of polymers at interfaces and to determine parameters that control adsorption or probe layer characteristics that are difficult to access otherwise. The idea is to have at the tip of an atomic force microscope (AFM), a probe chain and measure its behaviour near interfaces by pushing it to, or pulling it away from it. The self-consistent field modeling of this reveals that in the pulling mode—i.e., when the chain has an affinity for the surface—a typically inhomogeneous flower-like conformation forms with an adsorbed ’pancake’ and a stretched stem (tether) from the surface to the tip of the AFM. When about half the segments is in the tether it snaps loose in a first-order like fashion. The critical distance of the end-point from the surface and the critical force are experimentally accessible. Details of this transition depend on the surrounding of the test chain. Inversely, and this opens up many possibilities, the test chain reports about its surroundings. Our focus is on the classical case of homopolymers at interfaces. Pulling experiments may reveal the adsorption strength, the (average) chain length and/or the polymer concentration of the freely dispersed/adsorbed polymers. When the test-chain is non-adsorbing we envision that pushing this test-chain into the adsorption layer reports about various layer characteristics such as the layer thickness and (local) density. Moreover, when the test-chain has a length longer than the entanglement length, we can imagine that non-trivial dynamical properties of loops and tails may be scrutinised.

Polymer-mediated colloidal stability: on the transition between adsorption and depletion

Addition of polymers to a colloidal dispersion modulates the interactions between the colloids. We briefly review the effects of positive and negative adsorption (also termed depletion). The effective colloid-polymer interactions sensitively affect the colloidal phase behavior. We present a theoretical framework to predict the phase behavior of colloid-polymer mixtures for varying affinities between colloid and polymer, leading to either positive or negative adsorption of polymer segments. For certain conditions, polymers are neither depleted nor adsorbed: the polymer concentration is essentially constant up to the colloidal surface, a condition which we term neutral adsorption. Near this condition, the calculated phase diagrams reveal a stable-unstable-restabilisation transition with increasing polymer concentration. Similar effects have been reported experimentally, for instance as a function of temperature [Feng et al., Nat. Mat., 2015, 14, 61-65], which may modulate the effective polymer-colloid affinity. Understanding how to achieve neutral adsorption opens up the possibility of preparing highly dense, yet stable, colloid-polymer mixtures.

Sharp and fast: sensors and switches based on polymer brushes with adsorption-active minority chains

We propose a design for polymer-based sensors and switches with sharp switching transition and fast response time. The switching mechanism involves a radical change in the conformations of adsorption-active minority chains in a brush. Such transitions can be induced by a temperature change of only about ten degrees, and the characteristic time of the conformational change is less than a second. We present an analytical theory for these switches and support it by self-consistent field calculations and Brownian dynamics simulations.

Adsorption of a single polymer chain on a surface: effects of the potential range

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(g[perpendicular])(2)>/<R(g[parallel])(2)>=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.

Reconciling lattice and continuum models for polymers at interfaces

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 Δχ(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 Δχ(s)(c) is obtained by matching the boundary conditions in the two models. For depletion (positive c and Δχ(s)) the result is very simple: Δχ(s) = ln(1 + c/5). For adsorption (negative c and Δχ(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 Δχ(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 Δχ(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 Δχ(s)) from experiments. The relation Δχ(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.

Polymers at interfaces and in colloidal dispersions

This review is an extended version of the Overbeek lecture 2009, given at the occasion of the 23rd Conference of ECIS (European Colloid and Interface Society) in Antalya, where I received the fifth Overbeek Gold Medal awarded by ECIS. I first summarize the basics of numerical SF-SCF: the Scheutjens-Fleer version of Self-Consistent-Field theory for inhomogeneous systems, including polymer adsorption and depletion. The conformational statistics are taken from the (non-SCF) DiMarzio-Rubin lattice model for homopolymer adsorption, which enumerates the conformational details exactly by a discrete propagator for the endpoint distribution but does not account for polymer-solvent interaction and for the volume-filling constraint. SF-SCF corrects for this by adjusting the field such that it becomes self-consistent. The model can be generalized to more complex systems: polydispersity, brushes, random and block copolymers, polyelectrolytes, branching, surfactants, micelles, membranes, vesicles, wetting, etc. On a mean-field level the results are exact; the disadvantage is that only numerical data are obtained. Extensions to excluded-volume polymers are in progress. Analytical approximations for simple systems are based upon solving the Edwards diffusion equation. This equation is the continuum variant of the lattice propagator, but ignores the finite segment size (analogous to the Poisson-Boltzmann equation without a Stern layer). By using the discrete propagator for segments next to the surface as the boundary condition in the continuum model, the finite segment size can be introduced into the continuum description, like the ion size in the Stern-Poisson-Boltzmann model. In most cases a ground-state approximation is needed to find analytical solutions. In this way realistic analytical approximations for simple cases can be found, including depletion effects that occur in mixtures of colloids plus non-adsorbing polymers. In the final part of this review I discuss a generalization of the free-volume theory (FVT) for the phase behavior of colloids and non-adsorbing polymer. In FVT the polymer is considered to be ideal: the osmotic pressure Pi follows the Van 't Hoff law, the depletion thickness delta equals the radius of gyration. This restricts the validity of FVT to the so-called colloid limit (polymer much smaller than the colloids). We have been able to find simple analytical approximations for Pi and delta which account for non-ideality and include established results for the semidilute limit. So we could generalize FVT to GFVT, and can now also describe the so-called protein limit (polymer larger than the 'protein-like' colloids), where the binodal polymer concentrations scale in a simple way with the polymer/colloid size ratio. For an intermediate case (polymer size approximately colloid size) we could give a quantitative description of careful experimental data.

Polymer desorption under pulling: a dichotomic phase transition

The structural properties and phase behavior of a self-avoiding polymer chain on an adhesive substrate, subject to pulling at the chain end, are described by means of a grand canonical ensemble approach. We derive analytical expressions for the probability distributions of the basic structural units of an adsorbed polymer, such as loops, trains, and tails, in terms of the adhesive potential and applied pulling force f . In contrast to conventional, f=0 , polymer adsorption, the chain detachment transition under pulling turns out to be of first (rather than second) order, albeit it is dichotomic, i.e., no coexistence of different phase states exists. Also, the hitherto controversial value of the critical adsorption exponent varphi is found to depend essentially on the degree of interaction between different loops so that 0.34< or =varphi< or =0.59 . The theoretical predictions are verified by means of extensive Monte Carlo simulations.

Continuum formulation of the Scheutjens-Fleer lattice statistical theory for homopolymer adsorption from solution

Homopolymer adsorption from a dilute solution on an interacting (attractive) surface under static equilibrium conditions is studied in the framework of a Hamiltonian model. The model makes use of the density of chain ends n(1,e) and utilizes the concept of the propagator G describing conformational probabilities to locally define the polymer segment density or volume fraction phi; both n(1,e) and phi enter into the expression for the system free energy. The propagator G obeys the Edwards diffusion equation for walks in a self-consistent potential field. The equilibrium distribution of chain ends and, consequently, of chain conformational probabilities is found by minimizing the system free energy. This results in a set of model equations that constitute the exact continuum-space analog of the Scheutjens-Fleer (SF) lattice statistical theory for the adsorption of interacting chains. Since for distances too close to the surface the continuum formulation breaks down, the continuum model is here employed to describe the probability of chain configurations only for distances z greater than 2l, where l denotes the segment length, from the surface; instead, for distances z < or = 2l, the SF lattice model is utilized. Through this novel formulation, the lattice solution at z = 2l provides the boundary condition for the continuum model. The resulting hybrid (lattice for distances z < or = 2l, continuum for distances z > 2l) model is solved numerically through an efficient implementation of the pseudospectral collocation method. Representative results obtained with the new model and a direct application of the SF lattice model are extensively compared with each other and, in all cases studied, are found to be practically identical.

Liquid chromatography of polyoxyethylenes under critical conditions: A thermodynamic study

Polyethylene glycols (PEG), fatty alcohol ethoxylates (FAE) and fatty acid polyglycol esters were analyzed by liquid chromatography under critical conditions (LCCC) on typical reversed phase columns in mobile phases consisting of methanol-water and acetone-water in the region of the critical adsorption point (CAP) for the oxyethylene unit. Critical conditions can be adjusted by varying the mobile phase composition or temperature. In methanol-water the temperature dependence is almost negligible, while it is much more pronounced in acetone-water. Critical conditions for the oxyethylene unit were realized on octadecyl phases in methanol-water containing 85.8 wt.% methanol and in acetone-water at 89.6 wt.% acetone. In the van't Hoff plots the logarithm of the distribution coefficient was plotted versus the reciprocal temperature. In all plots straight lines were found, from slope and intercept of which the entropy and enthalpy changes were calculated. The relation of the thermodynamic parameters -deltaG/RT, -deltaH/RT, and deltaS/R to the numbers of A and B chain units in block copolymers AB and BAB at the CAP for A is studied both theoretically and experimentally. Various approaches for obtaining the critical temperature are discussed.

Influence of sequence correlations on the adsorption of random heteropolymers onto homogeneous planar surfaces

Using a reference system approach, we develop an analytical theory for the adsorption of random heteropolymers with exponentially decaying and/or oscillating sequence correlations on planar homogeneous surfaces. We obtain a simple equation for the adsorption--desorption transition line. This result as well as the validity of the reference system approach is tested by a comparison with numerical lattice calculations.

Theory of liquid chromatography of mono- and difunctional macromolecules. I. Studies in the critical interaction mode

The theory of liquid chromatography of mono- and difunctional polymers based on the model of ideal polymer chain in wide slit-like pores is presented. Analytical equations describing chromatographic behavior of functional macromolecules in both adsorption, exclusion and critical modes are derived and compared with experiments. The focus of this experimental study was on the verification of the theory in chromatography at critical conditions. Chromatographic behavior of low molar mass end-functionalized polyethylene glycols was found to be in a very good qualitative and in a reasonable quantitative agreement with the theory.

Coil-globule transition for regular, random, and specially designed copolymers: Monte Carlo simulation and self-consistent field theory

The coil-globule transition has been studied for A-B copolymer chains both by means of lattice Monte Carlo (MC) simulations using bond fluctuation algorithm and by a numerical self-consistent-field (SCF) method. Copolymer chains of fixed length with A and B monomeric units with regular, random, and specially designed (proteinlike) primary sequences have been investigated. The dependence of the transition temperature on the AB sequence has been analyzed. A proteinlike copolymer is more stable than a copolymer with statistically random sequence. The transition is more sharp for random copolymers. It is found that there exists a temperature below which the chain appears to be in the lowest energy state (ground state). Both for random and proteinlike sequences and for regular copolymers with a relatively long repeating block, a molten globule regime is found between the ground state temperature and the transition temperature. For regular block copolymers the transition temperature increases with block size. Qualitatively, the results from both methods are in agreement. Differences between the methods result from approximations in the SCF theory and equilibration problems in MC simulations. The two methods are thus complementary.