Scheutjens-Fleer Self-consistent Field Research Articles

Distribution of block copolymers in drying polymer films

HypothesisBlock copolymers (BCP) consisting of a polar block and a surface active apolar block are widely used for surface functionalization of polymer films. The characteristics of the copolymer blocks determine whether surface segregation and/or phase separation occurs, for a given bulk mixture. This data can be used to find the optimal BCP composition where high surface enrichment is obtained without accumulation of phase separated BCP in the bulk. MethodsThe distribution of poly(ethylene oxide)-polydimethylsiloxane (PEO-PDMS) BCP in a polymer formulation relevant for coating applications is systematically investigated. The surface segregation is studied in liquid formulations with surface tension measurements and dried films with X-ray photoelectron spectroscopy (XPS), whereas phase separation is quantified using turbidity measurements. The results are compared with Scheutjens-Fleer self-consistent field (SF-SCF) computations, which are also applied to determine the effect of film drying on BCP phase stability and surface segregation. FindingsLonger PDMS blocks result in lower interfacial tension of the liquid polymer mixture, whereas for the cured films, the largest PDMS concentration at the interface was obtained for intermediate PDMS block lengths. This is explained by the observation that phase separation already occurs at very low BCP concentrations for long PDMS blocks. The SCF predictions qualitatively agree with the experimental results and reveal that the BCP distribution changes significantly during film drying.

The Structure of Dipolar Polymer Brushes and Their Interaction in the Melt. Impact of Chain Stiffness.

By using the numerical lattice Scheutjens–Fleer self-consistent field (SF-SCF) method we have studied the effect of the restricted flexibility of grafted chains on the structure and mutual interaction of two opposing planar conventional and A-type dipolar brushes. Brushes are immersed in the solvent consisting of chains similar to the grafted ones. The increase of the chain rigidity enhances the segregation of grafted chains in a A-type dipolar brush into two populations: backfolded chains with terminal monomers near the grafting surface and chains with the ends at the brush periphery. The fraction of backfolded chains grows by an increase of the Kuhn segment length. It is shown that two opposite A-type dipolar brushes from semi-rigid chains are attracted to each other at short distances. The attraction becomes more pronounced and begins at larger distances for more rigid chains with the same brush characteristics: polymerization degree, grafting density, and dipole moments of monomer units. This attraction is connected with the dipole-dipole interactions between chains of oncoming brushes with oppositely directed dipoles penetrating deeply into each other upon contact. This effect of the chain rigidity is opposite to that for conventional brushes without dipoles in the chains. For such brushes, an increase in the chain rigidity leads to the enhanced repulsion between them.

Dual responsive PMEEECL-PAE block copolymers: a computational self-assembly and doxorubicin uptake study.

The self-assembly behaviour of dual-responsive block copolymers and their ability to solubilize the anticancer drug doxorubicin (DOX) has been investigated using all-atom molecular dynamics (MD) simulations, MARTINI coarse-grained (CG) force field simulation and Scheutjens–Fleer self-consistent field (SCF) computations. These diblock copolymers, composed of poly{γ-2-[2-(2-methoxyethoxy)ethoxy]ethoxy-ε-caprolactone} (PMEEECL) and poly(β-amino ester) (PAE) are dual-responsive: the PMEEECL block is thermoresponsive (becomes insoluble above a certain temperature), while the PAE block is pH-responsive (becomes soluble below a certain pH). Three MEEECL20–AEM compositions with M = 5, 10, and 15, have been studied. All-atom MD simulations have been performed to calculate the coil-to-globule transition temperature (Tcg) of these copolymers and finding appropriate CG mapping for both PMEEECL–PAE and DOX. The output of the MARTINI CG simulations is in agreement with SCF predictions. The results show that DOX is solubilized with high efficiency (75–80%) at different concentrations inside the PMEEECL–PAE micelles, although, interestingly, the loading efficiency is reduced by increasing the drug concentration. The non-bonded interaction energy and the RDF between DOX and water beads confirm this result. Finally, MD simulations and SCF computations reveal that the responsive behaviour of PMEEECL–PAE self-assembled structures take place at temperature and pH ranges appropriate for drug delivery.

Design principles for metamorphic block copolymer assemblies.

Certain block copolymer assemblies in selective solvents undergo dynamic morphology transitions (metamorphism) on varying the solution temperature. Despite the great application potential, there is a lack of fundamental understanding of the relationship between copolymer composition and the thermally-induced metamorphic behavior. Herein this relationship is studied by applying Scheutjens-Fleer Self-Consistent Field (SF-SCF) theory to develop fundamental design principles for thermoresponsive diblock copolymers exhibiting metamorphic behavior. It is found that metamorphism is caused by variation in the degree of stretching of the lyophobic blocks in response to changes in solvency. An optimal lyophobic/lyophilic block length ratio interval 3.5 ⪅ fB ⪅ 5.5 is identified. Such a fB window allows switching between spheres, cylinders and vesicles as preferred morphologies, with relatively small changes in the lyophobic block solvency. The transition from spheres to cylinders and from cylinders to bilayers can be controlled by varying fB, the overall degree of polymerization of the diblock copolymer, and by choosing an appropriate lyophilic block. Empirical relationships are provided to establish a connection between the SCF-SCF predictions and experimental observations.

Intramolecular micellization and nanopatterning in pH- and thermo-responsive molecular brushes.

Conformational transitions and nanoscale self-organization triggered in double pH- and thermo-responsive molecular brushes by varying environmental conditions are studied by means of analytical mean-field theory and numerical Scheutjens-Fleer self-consistent field modelling. Such molecular brushes are composed of multiple thermo-responsive side chains end-grafted onto the main chain (backbone) and are capable of acquiring ionic charges via reversible (de)protonation of the monomer units. Competition of long-range Coulomb repulsion with short-range solvophobic interactions leads to complex patterns in the intramolecular self-organization of molecular brushes. In particular, we observed formation of pearl necklace-like structures with multiple dense nanodomains formed by weakly ionized collapsed side chains and stabilized by a fraction protruding into the solution and strongly ionized ones. Such structures are thermodynamically stable in a certain parameter range and can be termed as intramolecular micelles. The stimuli-induced intramolecular nanopatterning occurs via a sequence of quasi-first order phase transitions corresponding to splitting/fusion of collapsed domains accompanied by jumps in the average degree of ionization and macromolecular dimensions. A re-entrant sequence of transitions is observed when the salt concentration is used as a control parameter. These theoretical predictions provide guidelines for design of smart unimolecular devices, for example multicompartment nanocarriers of active substances or nanosensors.

Controlling the Spatial Distribution of Solubilized Compounds within Copolymer Micelles.

The solubilization of lyophobic compounds in block copolymer micelles has been extensively investigated but remains only partially understood. There is a need to understand the fundamental parameters that determine the spatial distribution of the solubilized compounds within the micelles. Controlling this feature is a key aspect in the design of drug delivery systems with tailored release properties. Using Scheutjens–Fleer self-consistent field (SF-SCF) computations, we found that solubilization is regulated by a complex interplay between enthalpic and entropic contributions and that the spatial distribution can be controlled by the concentration and solubility of the guest compound in the dispersion medium. Upon solubilization, a characteristic change in size and mass of the micelles is predicted. This can be used as a fingerprint to indirectly assess the spatial distribution. Based on these findings, we developed two experimental protocols to control and assess the spatial distribution of lyophobic compounds within block copolymer micelles.

Temperature-Induced Re-Entrant Morphological Transitions in Block-Copolymer Micelles.

Using a combination of a mean-field theoretical method and the numerical Scheutjens-Fleer self-consistent field approach, we predict that it is possible to have re-entrant morphological transitions in nanostructures of diblock copolymers upon variation in temperature-mediated solubility of the associating blocks. This peculiar effect is explained by the different rates in variation of the density of the collapsed core domains and the corresponding interfacial energy as a function of the temperature. The theoretical findings are supported by existing experimental observations of reversed sequences of the morphological transitions occurring upon temperature variation in solutions of amphiphilic block copolymers.

A Hybrid Monte Carlo Self-Consistent Field Model of Physical Gels of Telechelic Polymers.

We developed a hybrid Monte Carlo self-consistent field technique to model physical gels composed of ABA triblock copolymers and gain insight into the structure and interactions in such gels. The associative A blocks of the polymers are confined to small volumes called nodes, while the B block can move freely as long as it is connected to the A blocks. A Monte Carlo algorithm is used to sample the node configurations on a lattice, and Scheutjens–Fleer self-consistent field (SF-SCF) equations are used to determine the change in free energy. The advantage of this approach over more coarse grained methods is that we do not need to predefine an interaction potential between the nodes. Using this MC-SCF hybrid simulation, we determined the radial distribution functions of the nodes and structure factors and osmotic compressibilities of the gels. For a high number of polymers per node and a solvent-B Flory–Huggins interaction parameter of 0.5, phase separation is predicted. Because of limitations in the simulation volume, we did however not establish the full phase diagram. For comparison, we performed some coarse-grained MC simulations in which the nodes are modeled as single particles with pair potentials extracted from SF-SCF calculations. At intermediate concentrations, these simulations gave qualitatively similar results as the MC-SCF hybrid. However, at relatively low and high polymer volume fractions, the structure of the coarse-grained gels is significantly different because higher-order interactions between the nodes are not accounted for. Finally, we compare the predictions of the MC-SCF simulations with experimental and modeling data on telechelic polymer networks from literature.

Interfacial Tension of Phase-Separated Polydisperse Mixed Polymer Solutions.

Aqueous two-phase systems provide oil-free alternatives in the formulation of emulsions in food and other applications. Theoretical interpretation of measurements on such systems, however, is complicated by the high polydispersity of the polymers. Here, phase diagrams of demixing and interfacial tensions are determined for aqueous solutions of two large polymers present in a mass ratio of 1:1, dextran (70 kDa) and nongelling gelatin (100 kDa), with or without further addition of smaller dextran molecules (20 kDa). Both in experiments and in calculations from Scheutjens–Fleer self-consistent field lattice theory, we find that small polymers decrease the interfacial tension at equal tie-line length in the phase diagram. After identifying the partial contributions of all chemical components to the interfacial tension, we conclude that excess water at the interface is partially displaced by small polymer molecules. An interpretation in terms of the Gibbs adsorption equation provides an instructive way to describe effects of polydispersity on the interfacial tension of demixed polymer solutions.

Dendron and Hyperbranched Polymer Brushes in Good and Poor Solvents.

We present a theory of conformational transition triggered by inferior solvent strength in brushes formed by dendritically branched macromolecules tethered to planar, concave, or convex cylindrical and spherical surfaces. In the regime of linear elasticity for brush-forming dendrons, an analytical strong stretching self-consistent field (SS-SCF) approach provides brush conformational properties as a function of solvent strength. A boxlike model is applied to describe the collapse transition in brushes formed by macromolecules with arbitrary treelike topology, including hyperbranched polymers. We demonstrate that an increase in the degree of branching, that is, an increase in the number of generations or/and functionality of branching points in tethered macromolecules, makes the swelling-to-collapse transition less sharp. A decrease in surface curvature has a similar effect. The numerical Scheutjens-Fleer self-consistent field approach is used to analyze the collapse transition in dendron brushes in the nonlinear stretching regime. It is demonstrated that inferior solvent strength suppresses stratification that is exhibited under good solvent conditions by densely grafted dendron brushes.

Complex coacervates formed across liquid interfaces: A self-consistent field analysis

The Scheutjens–Fleer self-consistent field (SF-SCF) theory is used to study complexation between two oppositely charged polyelectrolytes across an interface formed by two solvents, here called oil and water. The focus is on the composition and the lateral stability of such interfacial coacervate. One polyelectrolyte is chosen to be oil soluble and the other one prefers water, whereas the counter and salt ions are taken to distribute ideally over all phases. There exists an electrostatic associative driving force for the formation of the coacervate phase which increases with decreasing ionic strength and may be assisted by some specific affinity between the associating units and an effective poor solvency for the coacervate. As with respect to the lateral stability an unusual wetting scenario, called pseudo-partial wetting, presents itself, which results from interactions on two different length scales. On the segmental length the screening of oil–water contacts promotes the wetting by the coacervate: a pre-wetting jump-like transition takes place off-coexistence from a microscopically thin to a mesoscopically thin film. Usually this implies complete wetting. However, the mesoscopically thin film is exposed to long-ranged attractive electrostatic interactions and therefore cannot grow to macroscopic dimensions upon approach towards coexistence. Hence the system remains partial wet. The bulk correlation length controls the thickness of the mesoscopically thin film and as a result the wetting transition occurs extremely close to the bulk critical point. We therefore expect that a thick coacervate film typically is laterally inhomogeneous: there are drops on top of a mesoscopically thin coacervate film. This conclusion qualitatively explains the experimental observation that such a coacervate film scatters visible light.

Structure of Mixed Brushes Made of Arm-Grafted Polymer Stars and Linear Chains

The structure of a mixed brush made of arm-grafted polymer stars and grafted linear chains is investigated using the Scheutjens–Fleer self-consistent field method. It is shown that the mixing of stars and chains is thermodynamically favorable with respect to their lateral segregation in the brush. On the other hand, a segregation of linear and starlike macromolecules in the direction perpendicular to the grafting surface is observed. Conformations of stars and linear chains in the brush are determined by the overall molecular weight and the longest path length of linear and starlike macromolecules. Short linear chains occupy the space adjacent to the grafting surface and push the stars toward the brush periphery. In the case of long chains the interior of the brush is filled by the stars while the chains pass through the layer of the stars and expose their ends at the brush periphery. The most interesting is the intermediate situation where the linear chains have larger longest path but lower molecular we...

Persistence length of dendronized polymers: the self-consistent field theory.

We present numerical results for the thermodynamic rigidity and induced persistence length of dendronized polymers with systematically varied topology of their grafts obtained by the Scheutjens-Fleer self-consistent field method. The results were compared to predictions of an analytical mean-field theory. The two approaches have marked different predictions. In particular, the analytical theory predicts that the induced persistence length and the effective segment aspect ratio of dendronized polymers are increasing functions of the degree of branching of their side chains, whereas numerical calculations provide evidence of the opposite dependences. This discrepancy is argued to be due to the ability of side chains to repartition from the compressed to the dilated regions of a curved bottle brush, which is accounted for by the numerical, but not by the analytical method. The difference is most crucial in the light of the expected ability of dendronized polymers to have a liquid crystalline ordering in semi-dilute solutions.

Formation of a "Hollow" Interior in the Fourth-Generation Dendrimer with Attached Oligomeric Terminal Segments.

By using the Scheutjens-Fleer self-consistent field approach, the structure of the fourth-generation dendrimer with attached terminal chemically different oligomeric segments is studied theoretically. It is demonstrated that an incompatibility of terminal segments with inner dendrimer units leads to formation of a "hollow" core with reduced polymer density in the dendrimer center. This effect is enhanced with a deterioration in the solvent quality for terminal segments. This observation is in accordance with experimental results and molecular dynamics simulation data for an analogous system. It is established that the main factor determining the hollow core formation is the segregation between inner and terminal units because the main driving force for the effect is the localization of the terminal segments at the dendrimer periphery. The influence of structural parameters of dendrimer such as the number of generations and length of the terminal chain on this effect is also studied.

Particles Decorated by an Ionizable Thermoresponsive Polymer Brush in Water: Experiments and Self-Consistent Field Modeling

We have synthesized anionic multistimuli responsive core-shell polymer nanoparticles with low size dispersity composed of glassy poly(methyl methacrylate) (PMMA) cores of ca. 40 nm radius and poly(N-isopropylacrylamide) (PNIPAM) anionic brush-like shells with methacrylic acid comonomers. Using dynamic light scattering, we observed a volume phase transition upon an increase in temperature and this response was pH and ionic strength dependent. Already at room temperature we observed a pronounced polyelectrolyte effect, that is, a shift of the apparent pKa extracted from the degree of dissociation of the acids as a function of the pH. The multiresponsive behavior of the hydrophobic polyelectrolyte brush has been modeled using the Scheutjens-Fleer self-consistent field (SF-SCF) approach. Using a phenomenological relation between the Flory-Huggins χ parameter and the temperature, we confront the predicted change in the brush height with the observed change of the hydrodynamic radius and degree of dissociation and obtain estimates for the average chain lengths (number of Kuhn segments) of the corona chains, the grafting density and charge density distributions. The theory reveals a rich internal structure of the hydrophobic polyelectrolyte brush, especially near the collapse transition, where we find a microphase segregated structure. Considering this complexity, it is fair to state that the theoretical predictions follow the experimental data semiquantitatively, and it is attractive to attribute the observed disparity between theory and experiments to the unknown polydispersity of the chains, the unknown distribution of the charges, or other experimental complications. More likely, however, the deviations point to significant problems of the mean field theory, which focuses solely on the radial distributions and ignores the possibility of the formation of lateral (local) inhomogeneities in partially collapsed polyelectrolyte brushes. We argue that the PNIPAM brush at room temperature is already behaving nonideally.

Dendritic Spherical Polymer Brushes: Theory and Self-Consistent Field Modeling

Equilibrium structural properties of polymer brushes formed by dendrons grafted via the root segment onto spherical surfaces (dendritic spherical polymer brushes, DSPB) are studied by means of the Scheutjens–Fleer self-consistent field (SF-SCF) numerical approach and scaling analysis. In particular, we focus on the effects of the variable curvature of the surface on the polymer volume fraction distribution and extension of the individual dendrons in DSPB. A systematic comparison with spherical polymer brushes formed by linear polymer chains (LSPB) end-grafted to the surfaces of the same curvature radii is performed. We demonstrate that the difference in internal structural organization of DSPB and LSPB is most pronounced at small curvature radius of the grafting surface. In particular, the radial distribution of polymer volume fraction in DSPB is close to uniform, whereas in LSPB it decays in the radial direction following a power law. The quasi-plateau polymer volume fraction distribution in DSPB is ensu...

Self-consistent field predictions for quenched spherical biocompatible triblock copolymer micelles

We have used the Scheutjens–Fleer self-consistent field (SF-SCF) method to predict the self-assembly of triblock copolymers with a solvophilic middle block and sufficiently long solvophobic outer blocks. We model copolymers consisting of polyethylene oxide (PEO) as the solvophilic block and poly(lactic-co-glycolic) acid (PLGA) or poly(e-caprolactone) (PCL) as the solvophobic block. These copolymers form structurally quenched spherical micelles provided the solvophilic block is long enough. Predictions are calibrated on experimental data for micelles composed of PCL-PEO-PCL and PLGA-PEO-PLGA triblock copolymers prepared via the nanoprecipitation method. We establish effective interaction parameters that enable us to predict various micelle properties such as the hydrodynamic size, the aggregation number and the loading capacity of the micelles for hydrophobic species that are consistent with experimental findings.

On the Two-Population Structure of Brushes Made of Arm-Grafted Polymer Stars

The structure of a brush made of arm-grafted polymer stars is investigated using the Scheutjens–Fleer self-consistent field method. By using the “probe macromolecule” approach, conditional distributions of end- and branching points were obtained, which allowed for a detailed analysis of intramolecular correlations in the brush. Results strongly support a previously suggested “two population” picture of the star structure: stars in the brush are divided into two populations (i) those with weakly extended arms and (ii) those with a very strongly stretched grafting arm (stem) and all free arms extended toward the solvent. The stars in the “stretched” population have no arms that fold back toward the grafting surface and their free arms form a new sub-brush with effective grafting density determined solely by the total surface loading: molecular mass of polymer grafted onto unit area. With increasing grafting density or/and number of star arms the fraction of stars in the “stretched” population grows. The degree of backfolding of arms is estimated, the total fraction of backfolded arms is invariably small and decreases with increasing grafting density; only stars of the weakly extended population contribute to backfolding. The free ends within a given star are correlated: the ends of different arms are located approximately at the same distance from the grafting plane. Comparison of conditional ends distributions at fixed position of the branching points at different grafting densities reveals a conformational universality of grafted stars belonging to the weakly stretched population: the shape of the ends distribution does not depend on the grafting density but is determined solely by the position of the branching point. We also present analytical arguments showing that this effect is due to universal parabolic self-consistent potential acting on monomer units in the star brush, this potential is independent of the grafting density and the number of star arms.

Collapse of Polyelectrolyte Star. Theory and Modeling

The collapse of hydrophobic polyelectrolyte stars in aqueous solutions is studied using the Scheutjens–Fleer self-consistent field (SF-SCF) approach. The hydrophobic property of the segments tends to compact the stars, whereas the presence of charges has the opposite effect. As a result, star conformations can be switched from an extended, strongly hydrated, and swollen state to a collapsed state via semicollapsed, quasi-micellar state using control parameters such as the solvent quality, specified by the Flory–Huggins parameter, the pH value or the ionic strength. More specifically, there exists a range of parameters wherein the stars have an inhomogeneous radial structure with a collapsed region, referred to as the core, and a swollen region forming the corona. In such microphase segregated state the fraction of arms of the star that form the core, or alternatively escape into the swollen corona, can be controlled. The SF-SCF analysis is complemented with analytical models to rationalize the complex phase behavior.

Dendritic versus Linear Polymer Brushes: Self-Consistent Field Modeling, Scaling Theory, and Experiments

Equilibrium structural properties of polymer brushes formed by dendritic polymer chains (dendrons) are studied by means of Scheutjens−Fleer self-consistent field (SF-SCF) modeling and scaling analysis. Limiting cases of minimal and maximal possible losses of conformational entropy corresponding to different assumptions concerning distribution of elastic tension in the end-grafted dendrons are analyzed on the basis of the Flory-type scaling approach. The numerical SCF modeling indicates that the effective exponent of the power-law dependence for the height of dendritic brush on the grafting density differs from that derived within the Flory-type approximation. This is explained by changing of the intramolecular elastic tension distribution upon an increase in grafting density. The distributions of end and branching points are wide and exhibit multiple maxima, pointing to a broad distribution in the chain stretching. This distribution leads to monotonically decreasing overall density profiles. The theoretical results are in line with experimental findings on linear and dendritic poly(ethylene glycol) layers end-grafted onto TiO2 surfaces.