Compression and interpenetration of adsorption-active brushes.

Compression and interpenetration of two opposing polymer brushes formed by end-grafted adsorption-active chains are studied by the numerical self-consistent field approach and by analytical theory. For sufficiently strong polymer-surface attraction, a fraction of chains in the adsorption-active brush condenses into a near-surface layer, while the remaining ones form the outer brush with reduced effective grafting density. Analysis shows that the normal pressure in adsorption-active brushes can be understood in terms of the effective grafting density concept although the pressure at small separations is affected by the presence of the dense adsorbed phase. We propose a simple theory modification that accounts for this effect. We also formulate a procedure for extracting the value of the effective grafting density directly from the pressure vs separation curves by inverting the equation of state. In contrast to the normal pressure, the interpenetration of the two opposing adsorption-active brushes demonstrates a much more intricate behavior. At weak to moderate compressions, the effective grafting density concept works well but fails spectacularly at small interbrush separations. We identify two interpenetration regimes for adsorption-active brushes: (i) at separations larger than the ideal Gaussian coil size N1/2, the overlap of the two brushes is concentrated in the mid-plane region, in the same way as in brushes grafted onto non-attractive surfaces; (ii) at separations less than N1/2, the brush overlap is strongly enhanced in the wall regions where the attractive interaction plays an important role both in generating the dense layer for the "proper" brush and in attracting the "foreign" chains.

Adsorption-active polydisperse brush with tunable molecular mass distribution.

Recently, a novel class of responsive uncharged polymer brushes has been proposed [Klushin et al., J. Chem. Phys. 154(7), 074904 (2021)] where the brush-forming chains have an affinity to the substrate. For sufficiently strong surface interactions, a fraction of chains condenses into a near-surface layer, while the remaining ones form the outer brush with a reduced grafting density. The dense layer and the more tenuous outer brush can be seen as coexisting microphases. The effective grafting density of the outer brush is controlled by the adsorption strength and can be changed reversibly as a response to changes in environmental parameters. In this paper, we use numerical self-consistent field calculations to study this phenomenon in polydisperse brushes. Our results reveal an unexpected effect: Although all chains are chemically identical, shorter chains are adsorbed preferentially. Hence, with the increase in the surface affinity parameter, a reduction in the surface grafting density of the residual brush is accompanied by a change in the shape of its molecular mass distribution (MMD). In particular, an originally bidisperse brush can be effectively transformed into a nearly monodisperse one containing only the longer chain fraction. We introduce a method of assigning different chain conformations to one or the other microphase, based on analyzing tail length distributions. In a polydisperse brush with a uniform MMD, short chains are relegated to the adsorbed phase, leading to a narrower effective MMD in the residual brush. Preferential adsorption is not absolute, and longer chains are also partially involved in adsorption. As a result, not only the width of the distribution decreases but also its shape evolves away from the initial uniform distribution. We believe that the effect of preferential adsorption stems from a fundamental property of a polydisperse brush, which is characterized by a spectrum of chemical potential values for monomers belonging to chains of different lengths. Hence, preferential adsorption is also expected in polyelectrolyte brushes; moreover, brush polydispersity would affect coexistence with any other condensed phase, not necessarily related to adsorption.

Polymer brushes with reversibly tunable grafting density.

We propose a novel class of responsive polymer brushes, where the effective grafting density can be controlled by external stimuli. This is achieved by using end-grafted polymer chains that have an affinity to the substrate. For sufficiently strong surface interactions, a fraction of chains condenses into a near-surface layer, while the remaining ones form the outer brush. The dense layer and the more tenuous outer brush can be seen as coexisting microphases. The effective grafting density of the outer brush is controlled by the adsorption strength and can be changed reversibly and in a controlled way as a response to changes in environmental parameters. The effect is demonstrated by numerical self-consistent field calculations and analyzed by scaling arguments. Since the thickness of the denser layer is about a few monomer sizes, its capacity to form a microphase is limited by the product of the brush chain length and the grafting density. We explore the range of chain lengths and grafting densities where the effect is most pronounced. In this range, the SCF studies suggest that individual chains inside the brush show large rapid fluctuations between two states that are separated by only a small free energy barrier. The behavior of the brush as a whole, however, does not reflect these large fluctuations, and the effective grafting density varies smoothly as a function of the control parameters.

Phase transitions in single macromolecules: Loop-stretch transition versus loop adsorption transition in end-grafted polymer chains.

We use Brownian dynamics simulations and analytical theory to compare two prominent types of single molecule transitions. One is the adsorption transition of a loop (a chain with two ends bound to an attractive substrate) driven by an attraction parameter ε and the other is the loop-stretch transition in a chain with one end attached to a repulsive substrate, driven by an external end-force F applied to the free end. Specifically, we compare the behavior of the respective order parameters of the transitions, i.e., the mean number of surface contacts in the case of the adsorption transition and the mean position of the chain end in the case of the loop-stretch transition. Close to the transition points, both the static behavior and the dynamic behavior of chains with different length N are very well described by a scaling ansatz with the scaling parameters (ε - ε*)Nϕ (adsorption transition) and (F - F*)Nν (loop-stretch transition), respectively, where ϕ is the crossover exponent of the adsorption transition and ν is the Flory exponent. We show that both the loop-stretch and the loop adsorption transitions provide an exceptional opportunity to construct explicit analytical expressions for the crossover functions which perfectly describe all simulation results on static properties in the finite-size scaling regime. Explicit crossover functions are based on the ansatz for the analytical form of the order parameter distributions at the respective transition points. In contrast to the close similarity in equilibrium static behavior, the dynamic relaxation at the two transitions shows qualitative differences, especially in the strongly ordered regimes. This is attributed to the fact that the surface contact dynamics in a strongly adsorbed chain is governed by local processes, whereas the end height relaxation of a strongly stretched chain involves the full spectrum of Rouse modes.

Anomalous critical slowdown at a first order phase transition in single polymer chains.

Using Brownian dynamics, we study the dynamical behavior of a polymer grafted onto an adhesive surface close to the mechanically induced adsorption-stretching transition. Even though the transition is first order (in the infinite chain length limit, the stretching degree of the chain jumps discontinuously), the characteristic relaxation time is found to grow according to a power law as the transition point is approached. We present a dynamic effective interface model which reproduces these observations and provides an excellent quantitative description of the simulation data. The generic nature of the theoretical model suggests that the unconventional mixing of features that are characteristic for first-order transitions (a jump in an order parameter) and features that are characteristic of critical points (an anomalous slowdown) may be a common phenomenon in force-driven phase transitions of macromolecules.

Molecular weight effects on interfacial properties of linear and ring polymer melts: A molecular dynamics study.

Using molecular dynamics simulations, we study and compare the pressure, P, and the surface tension, γ, of linear chains and of ring polymers at the hard walls confining both melts into a slit. We examine the dependence of P and γ on the length (i.e., molecular weight) N of the macromolecules. For linear chains, we find that both pressure and surface tension are inversely proportional to the chain length, P(N)-P(N→∞)∝N-1,γ(N)-γ(N→∞)∝N-1, irrespective of whether the confining planes attract or repel the monomers. In contrast, for melts comprised of cyclic (ring) polymers, neither the pressure nor the surface tension is found to depend on molecular weight N for both kinds of wall-monomer interactions. While other structural properties as, e.g., the probability distributions of trains and loops at impenetrable walls appear quantitatively indistinguishable, we observe an amazing dissimilarity in the probability to find a chain end or a tagged monomer of a ring at a given distance from the wall in both kinds of polymeric melts. In particular, we demonstrate that the conformational equivalence of linear chains in a confined melt to a single chain under conditions of critical adsorption to a planar surface, established two decades ago, does also hold for ring polymers in a melt of linear chains. This analogy does not hold, however, for linear and ring chains in a confined melt of ring chains.

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.

Coil-bridge transition in a single polymer chain as an unconventional phase transition: theory and simulation.

The coil-bridge transition in a self-avoiding lattice chain with one end fixed at height H above the attractive planar surface is investigated by theory and Monte Carlo simulation. We focus on the details of the first-order phase transition between the coil state at large height H ⩾ Htr and a bridge state at H ⩽ Htr, where Htr corresponds to the coil-bridge transition point. The equilibrium properties of the chain were calculated using the Monte Carlo pruned-enriched Rosenbluth method in the moderate adsorption regime at (H/Na)tr ⩽ 0.27 where N is the number of monomer units of linear size a. An analytical theory of the coil-bridge transition for lattice chains with excluded volume interactions is presented in this regime. The theory provides an excellent quantitative description of numerical results at all heights, 10 ⩽ H/a ⩽ 320 and all chain lengths 40 < N < 2560 without free fitting parameters. A simple theory taking into account the effect of finite extensibility of the lattice chain in the strong adsorption regime at (H/Na)tr ⩾ 0.5 is presented. We discuss some unconventional properties of the coil-bridge transition: the absence of phase coexistence, two micro-phases involved in the bridge state, and abnormal behavior in the microcanonical ensemble.

Monofunctional polymers in liquid adsorption chromatography: determination of the interaction parameter in the range of weak interaction.

The main physical characteristics of monofunctionals in adsorption chromatography - the adsorption interaction parameter of the repeat units c and the interaction parameter of specific end group q - are discussed. Both parameters are independent on column dimensions and pore diameter, and depend on mobile phase composition. In a plot of elution volumes V(n) vs. the difference DeltaV=V(n)-V(n-1) in elution volumes of consecutive non-functional or monofunctional oligomers, straight lines with the same slope are obtained for sufficiently high molar masses. The intercept of these lines yield the accessible volumes of functionalized and non-functionalized oligomers. In the range of weak interaction, the interaction parameter of the repeat unit can be determined using monofunctional chains with strongly adsorbing end group. Scope and limitations of this approach are studied using monoalkyl ethers of polypropylene glycol as model polymers.

Universal properties of a single polymer chain in slit: Scaling versus molecular dynamics simulations.

We revisit the classical problem of a polymer confined in a slit in both of its static and dynamic aspects. We confirm a number of well known scaling predictions and analyze their range of validity by means of comprehensive molecular dynamics simulations using a coarse-grained bead-spring model of a flexible polymer chain. The normal and parallel components of the average end-to-end distance, mean radius of gyration and their distributions, the density profile, the force exerted on the slit walls, and the local bond orientation characteristics are obtained in slits of width D=4/10 (in units of the bead diameter) and for chain lengths N=50/300. We demonstrate that a wide range of static chain properties in normal direction can be described quantitatively by analytic model-independent expressions in perfect agreement with computer experiment. In particular, the observed profile of confinement-induced bond orientation is shown to closely match theory predictions. The anisotropy of confinement is found to be manifested most dramatically in the dynamic behavior of the polymer chain. We examine the relation between characteristic times for translational diffusion and lateral relaxation. It is demonstrated that the scaling predictions for lateral and normal relaxation times are in good agreement with our observations. A novel feature is the observed coupling of normal and lateral modes with two vastly different relaxation times. We show that the impact of grafting on lateral relaxation is equivalent to doubling the chain length.

Escape transition of a polymer chain from a nanotube: how to avoid spurious results by use of the force-biased pruned-enriched Rosenbluth algorithm.

A polymer chain containing N monomers confined in a finite cylindrical tube of diameter D grafted at a distance L from the open end of the tube may undergo a rather abrupt transition, where part of the chain escapes from the tube to form a "crownlike" coil outside of the tube. When this problem is studied by Monte Carlo simulation of self-avoiding walks on the simple cubic lattice applying a cylindrical confinement and using the standard pruned-enriched Rosenbluth method (PERM), one obtains spurious results, however, with increasing chain length the transition gets weaker and weaker, due to insufficient sampling of the "escaped" states, as a detailed analysis shows. In order to solve this problem, a new variant of a biased sequential sampling algorithm with resampling is proposed, force-biased PERM: the difficulty of sampling both phases in the region of the first order transition with the correct weights is treated by applying a force at the free end pulling it out of the tube. Different strengths of this force need to be used and reweighting techniques are applied. Using rather long chains (up to N=18000 ) and wide tubes (up to D=29 lattice spacings), the free energy of the chain, its end-to-end distance, the number of "imprisoned" monomers can be estimated, as well as the order parameter and its distribution. It is suggested that this algorithm should be useful for other problems involving state changes of polymers, where the different states belong to rather disjunct "valleys" in the phase space of the system.

What is the order of the two-dimensional polymer escape transition?

An end-grafted flexible polymer chain in three-dimensional space between two pistons undergoes an abrupt transition from a confined coil to a flowerlike conformation when the number of monomers in the chain, N, reaches a critical value. In two-dimensional (2D) geometry, excluded-volume interactions between monomers of a chain confined inside a strip of finite length 2L transform the coil conformation into a linear string of blobs. However, the blob picture raises questions about the nature of this escape transition. To check theoretical predictions based on the blob picture we study 2D single-polymer chains with excluded-volume interactions and with one end grafted in the middle of a strip of length 2L and width H by simulating self-avoiding walks on a square lattice with the pruned-enriched Rosenbluth method. We estimate the free energy, the end-to-end distance, the number of imprisoned monomers, the order parameter, and its distribution. It is shown that in the thermodynamic limit of large N and L but finite LN , there is a small but finite jump in several average characteristics, including the order parameter. We also present a theoretical description based on the Landau free energy approach, which is in good agreement with the simulation results. Both simulation results and the analytical theory indicate that the 2D escape transition is a weak first-order phase transition.

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 /=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.

Mechanical desorption of a single chain: unusual aspects of phase coexistence at a first-order transition.

The phase transition occurring when a single polymer chain adsorbed at a planar solid surface is mechanically desorbed is analyzed in two statistical ensembles. In the force ensemble, a constant force applied to the nongrafted end of the chain (that is grafted at its other end) is used as a given external control variable. In the z-ensemble, the displacement z of this nongrafted end from the surface is taken as the externally controlled variable. Basic thermodynamic parameters, such as the adsorption energy, exhibit a very different behavior as a function of these control parameters. In the thermodynamic limit of infinite chain length the desorption transition with the force as a control parameter clearly is discontinuous, while in the z-ensemble continuous variations are found. However, one should not be misled by a too-naive application of the Ehrenfest criterion to consider the transition as a continuous transition: rather, one traverses a two-phase coexistence region, where part of the chain is still adsorbed and the other part desorbed and stretched. Similarities with and differences from two-phase coexistence at vapor-liquid transitions are pointed out. The rounding of the singularities due to finite chain length is illustrated by exact calculations for the nonreversal random walk model on the simple cubic lattice. A new concept of local order parameter profiles for the description of the mechanical desorption of adsorbed polymers is suggested. This concept give evidence for both the existence of two-phase coexistence within single polymer chains for this transition and the anomalous character of this two-phase coexistence. Consequences for the proper interpretation of experiments performed in different ensembles are briefly mentioned.