Analytical Self-consistent Field Research Articles

Interpenetration and Friction of Brush-Coated Surfaces under External Load: Analytical Theory and SCF Calculations

We studied the compression and interpenetration properties of two opposing monodisperse polymer brushes under external load by analytical theory and by a numerical self-consistent field (SCF) approach. Analytical expressions are proposed for the brush density profiles in the interpenetration zone and verified by numerical SCF calculations. We quantify the interpenetration of two opposing brushes by the characteristic penetration length and by two integral parameters: the overlap integral, Γ, representing the number of interbush contacts, and the number of brush monomer units in the foreign half-space, Σ. The interpenetration parameters are studied in two conjugate ensembles as functions of the brush separation and of the external pressure. We propose a theoretical description of the solvent-mediated friction force in the low shearing rate regime on the basis of the Brinkman equation for two compressed brushes sliding against each other. We demonstrate that the total friction force which also includes direct brush–brush friction is expressed in terms of Γ and Σ. The SCF data for the interpenetration parameters Γ and Σ and hence for the total sliding friction force in the pressure ensemble collapse onto universal master curves when rescaled by the factor (N/σ)1/3 as suggested by the theory, where N is the chain length and σ is the surface grafting density. Finally, we define the kinetic friction coefficient as a function of the external pressure and analyze its universal rescaled behavior in the two limits of nondraining and free-draining brushes.

Unsynchronous conformational transitions induced by the asymmetric adsorption-response of an active diblock copolymer in an inert brush.

To exploit the chemical asymmetry of diblock copolymer chains on the design of high-performance switch sensors, we propose an analytically tractable model system which contains an adsorption-responsive diblock copolymer in an otherwise inert brush, and study its phase transitions by using both analytical theory and self-consistent field calculations. The copolymer chain is chemically asymmetric in the sense that the two blocks assume different adsorption strengths, which is characterized by the defined adsorption ratio. We found that the conformation states, the number of stable phases, and transition types are mainly controlled by the length of each block and the adsorption ratio. In particular, when the length of the ungrafted block is longer than the brush chains, and the adsorption ratio is smaller than a critical value, the copolymer chain shows three thermodynamically stable states, and undergoes two unsynchronous transitions, where the two blocks respond to the adsorption in a different manner, when the adsorption changes from weak to sufficiently strong. For this kind of three-state transition, the transition point, transition barrier, and transition width are evaluated by using the self-consistent field method, and their scaling relationship with respect to the system parameters is extracted, which matches reasonably well with the predictions from the analytical theory. The self-consistent field calculations also indicate that the conformational transitions involved in the three-state transition process are sharp with a low energy barrier, and interestingly, barrier-free transitions are observed. Our finding shows that the three-state transitions not only specify a region where high performance unsynchronous switch sensors can be exploited, but may also provide a useful model understanding the unsynchronous biological processes.

Analytic Linear Vibronic Coupling Method for First-Principles Spin-Dynamics Calculations in Single-Molecule Magnets.

Accurate modeling of vibronically driven magnetic relaxation from ab initio calculations is of paramount importance to the design of next-generation single-molecule magnets (SMMs). Previous theoretical studies have been relying on numerical differentiation to obtain spin-phonon couplings in the form of derivatives of spin Hamiltonian parameters. In this work, we introduce a novel approach to obtain these derivatives fully analytically by combining the linear vibronic coupling (LVC) approach with analytic complete active space self-consistent field derivatives and nonadiabatic couplings computed at the equilibrium geometry with a single electronic structure calculation. We apply our analytic approach to the computation of Orbach and Raman relaxation rates for a bis-cyclobutadienyl Dy(III) sandwich complex in the frozen-solution phase, where the solution environment is modeled by electrostatic multipole expansions, and benchmark our findings against results obtained using conventional numerical derivatives and a fully electronic description of the whole system. We demonstrate that our LVC approach exhibits high accuracy over a wide range of coupling strengths and enables significant computational savings due to its analytic, "single-shot" nature. Evidently, this offers great potential for advancing the simulation of a wide range of vibronic coupling phenomena in magnetism and spectroscopy, ultimately aiding the design of high-performance SMMs. Considering different environmental representations, we find that a point charge model shows the best agreement with the reference calculation, including the full electronic environment, but note that the main source of discrepancies observed in the magnetic relaxation rates originates from the approximate equilibrium electronic structure computed using the electrostatic environment models rather than from the couplings.

Effects of cosolvent partitioning on conformational transitions and tethered chain flexibility in spherical polymer brushes.

In this work, based on the framework of preferential adsorption concept and analytical self-consistent field (SCF) theory, a model is proposed to investigate the reentrant transition experimentally observed from the thermoresponsive spherical brush in a series of aqueous alcohol solutions. The interaction between monomers is incorporated into the model. Conformational transitions of the spherical brush are quantitatively correlated to the physical parameters, including the number of adsorbed cosolvents which facilitates the solvent quality, the number of cosolvent bridges which drives the brush collapse, as well as their partition coefficients between the brush and the bulk solution. An analytical formula for the number of Kuhn segments per tethered chain is obtained based on the analytical SCF theory, which elucidates the flexibility of tethered chains in the intricate system of multicomponents involving the conformational transitions. Under the experimental conditions associated with the cosolvent-brush interaction, the variation of the monomer chemical potential with the monomer concentration indicates that the monomer distribution of the spherical brush remains continuous. The analysis based on the SFC theory also reveals that the distribution of adsorbed cosolvents is a positive parabola while the distribution of cosolvent bridges appears to be an exponential decay function, implying that the intervening space between tethered chains, rather than the number of adsorbed cosolvents, plays a crucial role in forming the cosolvent bridge. We demonstrate that the model formulated for the reentrant transition under weaker cosolvent-brush interactions provides guidelines for the one under stronger nanoparticle-brush interactions.

Efficient implementation of the analytical second derivatives of hartree-fock and hybrid DFT energies within the framework of the conductor-like polarizable continuum model.

Calculation of vibrational frequencies for solvated systems is essential to study reactions in complex environments. In this paper, we report the implementation of the analytical self-consistent field Hessian at the Hartree-Fock and density functional theory levels in the framework of the conductor-like polarizable continuum model (C-PCM) into the ORCA quantum chemistry suite. The calculated vibrational frequencies agree very well with those computed through numerical differentiation of the analytical gradients. The deviation between both sets of data is smaller than 3 cm-1 for frequencies larger than 200 cm-1 and smaller than 5 cm-1 for the low-frequency regime (100 cm-1 < ω < 200 cm-1 ). The accuracy of the frequencies is not significantly affected by the size of the density functional theory (DFT) integration grid, with a deviation lower than 0.5 cm-1 between data computed with the smallest and that with the largest DFT grid size. The calculation of the analytical Hessian is between 3 and 12 times faster than its numerical counterpart. The C-PCM terms only add an overhead of 10-30% relative to the gas phase calculations. Finally, for acetone, the (B3LYP) values for the frequency shifts obtained in going from the gas phase to liquid acetone are in agreement with experiment. © 2019 Wiley Periodicals, Inc.

Structure and lubrication of solvent-free dendron brushes

Using analytical and numerical self-consistent field (SCF) methods we analyze equilibrium structure of dry (solvent-free) brush of branched polymers in wide range of chain grafting densities. A detailed analysis is performed for starlike polymers end-tethered to planar surface by terminal monomer of one of their branches. At moderate grafting densities ensuring Gaussian elasticity of tethered stars the brush structure is described by strong stretching SS-SCF theory employing parabolic molecular potential. At high grafting densities an approximate analytical AA-SCF theory is used to describe stratification of dry brush. Analytical predictions are supplemented by results of numerical Scheutjens-Fleer SF-SCF model. Interpenetration between dry brushes is analyzed as a function of grafting density and degree of branching of tethered chains. Comparison of interpenetration and friction between dry brushes with varied chain architectures pointed at optimal lubrication by macrocycles tethered to the surface by branching point. It is demonstrated that shear stress in sliding brushes of starlike macromolecules sharply decreases at high grafting densities corresponding to almost full stretching of the longest elastic path in the polymers.

Diamond-Forming Block Copolymers and Diamond-like Morphologies: A New Route toward Efficient Block Copolymer Membranes

A new promising class of ordered block copolymer morphologies in thin films we call diamond-like morphologies (DLM) is theoretically described in detail. This class comprises the cubic morphologies possessing diamond symmetry Fd3m as well as its tetragonal and orthorhombic generalizations. The characteristic property of the DLM is that the corresponding percolation cluster contains both perpendicular and parallel (with respect to the substrate) intertwining channels, which is expected to improve noticeably the permeability efficiency of such quasi-isotropic morphologies as compared to the anisotropic lamellar and cylindrical ones. On the basis of both the weak segregation theory (WST) analytical consideration and self-consistent field theory (SCFT) numerical procedure, we find relationships between the microscopic parameters of the ternary ABC symmetric block copolymers of a specified composition and their morphology in thin films. The effects of confinement (film width) and the film walls selectivity ar...

Theory of Brushes Formed by Ψ-Shaped Macromolecules at Solid-Liquid Interfaces.

We present a theoretical analysis targeted to describe the structural properties of brushes formed by Ψ-shaped macromolecules tethered by terminal segment of stem to planar surface while exposing multiple free branches to the surrounding solution. We use an analytical self-consistent field approach based on the strong stretching approximation, and the assumption of Gaussian elasticity for linear chain fragments of the tethered macromolecules. The effect of weak and strong polydispersity of branches is analyzed. In the case of weakly polydisperse macromolecules, variations in length of branches lead to a more uniform polymer density distribution with slight increase in the brush thickness compared to the case of monodisperse chains with the same degree of polymerization. We demonstrate that in contrast to linear chains, strong polydispersity of Ψ-shaped macromolecules does not necessarily lead to strong perturbations in polymer density distribution. In particular, mixed brushes of the so-called "mirror" dendrons (in which number of stem monomers in one component coincides with number of monomers in a branch of the other component, and vice versa) give rise to a unified polymer density distribution with shape independent of the brush composition. The predictions of analytical theory are systematically compared to the results of numerical self-consistent field modeling based on the Scheutjens-Fleer approach.

Self-consistent field model of brushes formed by root-tethered dendrons

We present an analytical self-consistent field (scf) theory that describes planar brushes formed by regularly branched root-tethered dendrons of the second and third generations. The developed approach gives the possibility for calculation of the scf molecular potential acting at monomers of the tethered chains. In the linear elasticity regime for stretched polymers, the molecular potential has a parabolic shape with the parameter k depending on architectural parameters of tethered macromolecules: polymerization degrees of spacers, branching functionalities, and number of generations. For dendrons of the second generation, we formulate a general equation for parameter k and analyze how variations in the architectural parameters of these dendrons affect the molecular potential. For dendrons of the third generation, an analytical expression for parameter k is available only for symmetric macromolecules with equal lengths of all spacers and equal branching functionalities in all generations. We analyze how the thickness of dendron brush in a good solvent is affected by variations in the chain architecture. Results of the developed scf theory are compared with predictions of boxlike scaling model. We demonstrate that in the limit of high branching functionalities, the results of both approaches become consistent if the value of exponent bin boxlike model is put to unity.In conclusion, we briefly discuss the systems to which the developed scf theory is applicable. These are: planar and concave spherical and cylindrical brushes under various solvent conditions (including solvent-free melted brushes) and brush-like layers of ionic (polyelectrolyte) dendrons.

Excluded volume effects in compressed polymer brushes: A density functional theory.

A classical density functional theory (DFT) is applied to investigate the behavior of compressed polymer brushes composed of hard-sphere chains. The excluded volume interactions among the chain segments are explicitly treated. Two compression systems are used to study the behavior of brush-wall and brush-brush interactions. For the brush-brush systems, an obvious interpenetration zone has been observed. The extent of the interpenetration depends strongly on the grafting density. Furthermore, the repulsive force between the brush and wall or between the two brushes has been obtained as a function of the compression distance. Compared to the prediction of the analytic self-consistent field theory, such force increases more rapidly in the brush-wall compression with high polymer grafting densities or at higher compressions. In the brush-brush compression system, the interpenetration between the two compressed brushes creates a "softer" interaction. The influence of hard-sphere solvents on the behavior of compressed brushes is also discussed.

Analytical self-consistent field model of arm-grafted starlike polymers in nonlinear elasticity regime

We develop approximate analytical (AA) model to describe the structure of a brush made of starlike polymers (regularly branched dendrons of the first generation) grafted by one arm onto a planar surface. It is demonstrated that introduction of two sub-layers with distinct populations of strongly and weakly stretched dendrons allows for adequate semi-analytical description of these end-grafted macromolecules in nonlinear elasticity regime. We calculate brush thickness, fraction of strongly stretched dendrons and monotonously decreasing polymer density profiles as a function of polymer grafting density and number of branches in starlike macromolecule. The predictions of AA model are compared with the results of numerical self-consistent field model of Sheutjens and Fleer (SF-SCF). Good agreement between the two models is found at high grafting densities and/or at large number of branches of starlike polymers.

Immersion energy and polymer-mediated depletion interactions between nanocolloids as studied by analytic self-consistent field theory

By applying the Edwards self-consistent field theory, we calculate the polymer density and free energy excesses caused by the presence of nanocolloids in the excluded volume polymer system. Using the obtained results, we have calculated the depletion potential U as a function of the separation between colloids, colloid radius, polymer volume fraction, and polymer gyration radius. Upon analyzing the obtained results against the known exact asymptotic small-separation limit and scaling relations, we propose an approximate expression for U that builds upon our exact result. This expression is shown to give excellent agreement with Monte Carlo simulations. Reliability of the derived analytical expression for the depletion potential and its relation to the previous theoretical and simulation work are thoroughly discussed.

A new numerical approach to dense polymer brushes and surface instabilities

We present a numerical self-consistent field (SCF) method which describes freely jointed chains of spherical monomers applied to densely grafted polymer brushes. We discuss both the Flory-Huggins model and the Carnahan-Starling equation of state and show the latter being preferable within our model at polymer volume fractions above 10%. We compare the results of our numerical method with data from molecular dynamics (MD) simulations [G.-L. He, H. Merlitz, J.-U. Sommer, and C.-X. Wu, Macromolecules 40, 6721 (2007)] and analytical SCF calculations [P. M. Biesheuvel, W. M. de Vos, and V. M. Amoskov, Macromolecules 41, 6254 (2008)] and obtain close agreement between the density profiles up to high grafting densities. In contrast to prior numerical and analytical studies of densely grafted polymer brushes our method provides detailed information about chain configurations including fluctuation, depletion, and packing effects. Using our model we could study the recently discovered instability of densely grafted polymer brushes with respect to slight variations of individual chain lengths, driven by fluctuation effects [H. Merlitz, G.-L. He, C.-X. Wu, and J.-U. Sommer, Macromolecules 41, 5070 (2008)]. The obtained results are in very close agreement with corresponding MD simulations.

Adsorption of Anionic Surfactants in a Nonionic Polymer Brush: Experiments, Comparison with Mean-Field Theory, and Implications for Brush−Particle Interaction

The adsorption of the anionic surfactants sodium dodecyl sulfate (SDS) and sodium dodecyl benzene sulfonate (SDBS) in poly(ethylene oxide) (PEO) brushes was studied using a fixed-angle optical flow-cell reflectometer. We show that, just as in solution, there is a critical association concentration (CAC) for the surfactants at which adsorption in the PEO brush starts. Above the critical micelle concentration (CMC) the adsorption is found to be completely reversible. At low brush density the adsorption per PEO monomer is equal to the adsorption of these surfactants in bulk solution. However, with increasing brush density, the number of adsorbed surfactant molecules per PEO monomer decreases rapidly. This decrease is explained in terms of excluded volume interactions plus electrostatic repulsion between the negatively charged surfactant micelles. Experimentally, a plateau value in the total adsorption is observed as a function of grafting density. The experimental results were compared to the results of an analytical self-consistent field (aSCF) model, and we found quantitative agreement. Additionally, the model predicts that the plateau value found is in fact a maximum. Both experiments and model calculations show that the adsorption scales directly with the polymerization degree of the polymers in the brush. They also show that an increase in the ionic strength leads to an increase in the adsorbed amount, which is explained as being due to a decrease in the electrostatic penalty for the adsorption of the SDS micelles. The adsorption of SDS micelles changes the interactions of the PEO brush with a silica particle. This is illustrated by atomic force microscopy (AFM) measurements of the pull-off force of a silica particle from a PEO brush: at high enough PEO densities, the addition of SDS leads to a very strong reduction in the force necessary to detach the colloidal silica particle from the PEO brush. We attribute this effect to the large amount of negative charge incorporated in the PEO brush due to SDS adsorption.

Nanostructure of the Interface Modified by Grafted Polymers: A Monte Carlo Simulation

The thermodynamic, conformational and orientational properties of polymer melts grafted on a solid substrate were obtained from a novel Monte Carlo (MC) simulation of coarse-grained model of polyethylene (PE). The interface between a non-interacting hard surface and a bulk PE melt, with all chains of which are grafted on the plane, has been studied. Different PE melts, of mean molecular length from C40 and C80, have been investigated, at grafting densities ranging from 0.92 to 1.85 nm. Profiles of monomer density and free end density, bond orientation, and average monomer position along a chain were studied. Quantitative measured in the simulations are derived from the analytical self-consistent field (SCF) theory and compared with the simulation data. The conformational and orientational properties can be quite accurately described by the theory, with some discrepancies observed near the wall and at the tail of the profile. Additional results concerning thermodynamic and surface energy of the brush are also presented.

Effect of end-group sticking energy on the properties of polymer brushes: Comparing experiment and theory

Using surface force balance measurements we have established that polystyrene chains bearing three zwitterionic groups have a higher end-group sticking energy than equivalent chains bearing a single zwitterionic group. In a good solvent, polystyrene chains end-functionalized with three zwitterionic groups form brushes of a higher surface coverage than those bearing a single zwitterion. The increase in surface coverage is slow compared with the initial formation of the brush. Measurements of the refractive index allow us to directly quantify the variation of surface coverage, permitting comparison with models for the kinetics of brush formation based on scaling theory and an analytical self-consistent field. We find qualitative support for associating the kinetic barrier with the energy required for an incoming chain to stretch as it penetrates the existing brush.

Annealed Star-Branched Polyelectrolytes in Solution

Equilibrium conformations of annealed star-branched polyelectrolytes (polyacids) are calculated with a numerical self-consistent-field (SCF) model. From the calculations we obtain also the size and charge of annealed polyelectrolyte stars as a function of the number of arms, pH, and the ionic strength. The results are compared with predictions from analytical theory. Upon varying the number of branches or the ionic strength of the solution, the star size changes nonmonotonically, which is in agreement with the analytical predictions. The salt concentration at this maximum is directly related to the charge density of the star. The internal structural properties of the star corona (the polymer density, the ionization profiles, and the distribution of the end points) are analyzed. The shape of the density profiles indicates increasing local stretching of the branches as a function of the distance from the star center. Furthermore, a bimodal end-point distribution is found and interpreted in analogy to that predicted earlier by analytical SCF theory for planar polyelectrolyte brushes. Results of recent experiments with annealed star-shaped micelles are discussed on the basis of our numerical model calculations.

Structural Study of Langmuir Monolayers Containing Lipidated Poly(ethylene glycol) and Peptides

The structure of Langmuir monolayers containing either a lipidated poly(ethylene glycol) (PEG-lipid) or a lipidated peptide (peptide-amphiphile) or a binary mixture of both was studied using neutron reflectivity. The PEG portion of the PEG-lipid extends into the water, forming dense polymer “brushes”. The PEG volume fraction profiles and the brush height were evaluated from the reflectivity curves for monolayers containing PEG-lipids with PEG molecular masses of 120, 750, 2000, and 5000 Da at various grafting densities. At relatively low surface densities, the segmental concentration profile for DSPE-PEG5000 (1,2-distearoyl-sn-glycero-3-phosphoethanolamine-N-[(ethylene glycol)n], DSPE) and DSPE-PEG2000 was well-described by the parabolic profile predicted by the analytical self-consistent mean field theory. An increase in the surface density produced “flattening” of the profile, which became more pronounced as the chain length decreased. The dependence of the brush height on the surface density and the ch...

Polyelectrolytes tethered to a similarly charged surface

Conformations of weakly charged quenched polyelectrolyte chains tethered to a similarly charged planar surface are analyzed on the basis of a combination of scaling, analytical, and numerical self-consistent field (SCF) approaches. Scaling theory predicts universal power law dependences of the large-scale conformational properties (like the end-to-end distance) of grafted chains on the overall surface charge per unit area. The SCF approach allows analysis of the detailed conformational structure of grafted polyions as a function of the distribution of immobilized charges between the surface and grafted chains. The analytical solution is only available in the limiting cases of sparse grafting of polyions to the charged plane and sufficiently dense grafting of polyions to a neutral surface. In the intermediate case when both interchain interactions and interaction of grafted chains with the surface are important, only numerical solutions can be obtained. We consider various ways to distribute the charges between the surface and the brush chains while keeping the sum of the two contributions constant. Upon increasing the charge on the surface, by concomitant reduction of the grafting density of the tethered chains we found: (i) an increase in height of the polymer layer; (ii) the development of a depletion zone of end points near the surface; and (iii) a sharpening of the end-point distribution with the peak shifting away from the surface. In the appropriate limiting cases excellent agreement of analytical SCF predictions is obtained with the numerical results.

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.