Monte Carlo investigations of dense copolymer systems, 1. Pivot algorithm and pair distribution function

ABA triblock copolymers (the solvent being a θ-solvent for B and an athermal one for A) as well as their constituent homopolymers were investigated by means of Monte Carlo simulations in a 5-choice cubic lattice. The polymer concentration (represented by the fraction ϕ of sites occupied by the polymer) varied from 0 (isolated chain) to 0.80. An investigation of the usefulness of the pivot algorithm in dense systems resulted in reasonable acceptance fractions up to a volume fraction of 0.12 (athermal chain and ABA copolymer) and 0.05 (θ-chain). For larger volume fractions only the chain-ends remain mobile. The overlap concentration of the polymers defined by several quantities was approximately 0.07 (athermal chain), 0.12 (θ-chain) and 0.09 (ABA copolymer). At a polymer fraction of 0.32, the chains had the same number of inter- and intramolecular contacts on average. At higher concentrations, the behaviour of the chains was primarily determined by intermolecular interactions. Contrary to isolated pairs, the pair distribution function g(r) of two athermal chains in a (cubic) box exceeded unity at intermediate distances because of the influence of the finite size of the box. The larger the size of the box, the smaller the (positive) deviation was. In the limit of r → 0 the pair distribution function g(r) – being smaller than unity – increased with increasing concentration while the maximum at intermediate distances simultaneously decreased. Ultimately, at the highest concentration, the pair distribution function resembled that of isolated θ-chains. The concentration dependence of the θ-pair distribution function itself, however, is negligible. At low concentrations the pair distribution function of the ABA triblock copolymer behaved like that of an athermal chain (up to ϕ ≈ 0.40) where characteristic oscillations around 1 developed. This might be taken as indicative of the formation of a lamellar phase.     

Phase behavior and structure formation in linear multiblock copolymer solutions by Monte Carlo simulation

The solution phase behavior of short, strictly alternating multiblock copolymers of type (A(n)B(n))(m) was studied using lattice Monte Carlo simulations. The polymer molecules were modeled as flexible chains in a monomeric solvent selective for block type A. The degree of block polymerization n and the number of diblock units per chain m were treated as variables. We show that within the regime of parameters accessible to our study, the thermodynamic phase transition type is dependent on the ratio of m / n. The simulations show microscopic phase separation into roughly spherical aggregates for m / n ratios less than a critical value and first-order macroscopic precipitation otherwise. In general, increasing m at fixed n, or n at fixed m, promotes the tendency toward macroscopic phase precipitation. The enthalpic driving force of phase change is found to universally scale with chain length for all multiblock systems considered and is independent of the existence of a true phase transition. For aggregate forming systems at low amphiphile concentrations, multiblock chains are shown to self-assemble into intramolecular, multichain clusters. Predictions for microstructural dimensions, including critical micelle concentration, equilibrium size, shape, aggregation parameters, and density distributions, are provided. At increasing amphiphile density, interaggregate bridging is shown to result in the formation of networked structures, leading to an eventual solution-gel transition. The gel is swollen and consists of highly interconnected aggregates of approximately spherical morphology. Qualitative agreement is found between experimentally observed physical property changes and phase transitions predicted by simulations. Thus, a potential application of the simulations is the design of multiblock copolymer systems which can be optimized with regard to solution phase behavior and ultimately physical and mechanical properties.     

Monte Carlo study of chain conformations in the swollen middle layer of onion‐skin polymeric micelles

Conformations of chains in swollen middle layers of onion‐skin micelles were studied by Monte Carlo simulations on a tetrahedral lattice under conditions that mimic real systems of swollen onion‐skin micelles. Polymer blocks are modeled as tethered self‐avoiding chains, enclosed in a narrow spherical layer. Average density of segments, 〈g_S〉 ca. 0.6, corresponds to swollen micellar systems. Only the excluded volume effect was taken into account since it plays the most important role in dense polymer systems. Individual chains are described by equivalent ellipsoids of gyration. Distributions of the ellipsoid half‐axes were calculated during simulations. Results based on a large series of simulations indicate that the middle layer‐forming blocks may be described as prolonged ellipsoids oriented preferentially perpendicular to the radial direction. Analysis of the data concerning the orientations of end‐to‐end vectors and distributions of segments within one chain indicates that individual chains are strongly interpenetrated and the multi‐chain system is fairly disordered.     

Molecular,Nanostructural and Mechanical Characteristics of Lamellar Triblock Copolymer Blends: Effects of Molecular Weight and Constraint

While theoretical and experimental efforts have thoroughly addressed microphase‐ordered AB diblock copolymer blends with a parent homopolymer (hA or hB) or a second block copolymer, surprisingly few studies have considered comparable ABA triblock copolymers in the presence of hB or an AB diblock copolymer. In this study, we elucidate the roles of additive molecular weight and constraint by examining three matched series of miscible ABA/hB and ABA/AB blends. Self‐consistent field theory is employed to analyze molecular characteristics, e. g., segmental distributions, microdomain periods and midblock bridging fractions, as functions of blend composition. Predictions are compared to morphological characteristics discerned by transmission electron microscopy and small‐angle X‐ray scattering. The corresponding mechanical properties of these blends are measured by dynamic mechanical analysis. The results of this comprehensive work reveal that addition of hB swells the B‐lamellae of the ABA copolymer and has a generally deleterious effect on both the dynamic elastic modulus and midblock bridging fraction. In contrast, addition of a lamellar or cylindrical AB copolymer to the same ABA copolymer can promote an increase or decrease in lamellar period and bridging fraction, depending on relative block sizes.     

A computer simulation study of the segregation of amphiphiles in binary immiscible matrices: short asymmetric copolymers in short homopolymers

The bulk and interfacial properties of ternary mixtures with asymmetric amphiphiles (A2B8) in A2 and B2 matrices and in A2 and B10 matrices are investigated by the dissipative particle dynamics type of molecular-dynamics simulations. The monomer concentrations of A2B8(phiA2B8) studied are below the critical micelle concentration (phiA2B8(cmc)) for the formation of micelles in the presence of an adsorbed amphiphilic monolayer at the interface. Macrophase separation from the mixed phase to the segregated state with A-rich and B-rich coexisting phases and the segregation of A2B8 at the interface are thermodynamically gradual but are accompanied by a pronounced stretching and orientation of the constituent chains. The segregation of A2B8 at the interface broadens the interfacial region and reduces the interfacial tension. The chain conformation of the asymmetric amphiphilic molecules and the interfacial properties are dominated by the majority block in the amphiphilic chain and dependent on the composition of the matrix in contact with the majority block. In the A2 and B2 matrices, the B8 blocks in A2B8 chains at the interface resemble a wet brush swollen by short B2 chains. Swelling is responsible for the pronounced stretching and orienting of the amphiphilic chains and the reduced interfacial amphiphile enrichment. At the same interfacial amphiphile excess, however, swollen amphiphiles are more efficient in reducing the interfacial tension than nonswollen amphiphiles.     

Bimodal polymer mushrooms: compressive forces and specificity toward receptor surfaces

End-grafted poly(ethylene glycol) (or PEG) polymer chains are used to extend the in vivo circulation time of targeted liposomes and nanoparticles; however, the most efficacious structure for also imparting high target specificity remains unknown. Using the surface force apparatus, we have measured the specific and nonspecific forces between bimodal mixtures of grafted polymer mushrooms and model receptor surfaces. Specifically, supported lipid membranes anchoring 2000 or 5000 Da PEG with a controlled fraction of PEG(2000) bearing biotin ligands were compressed against opposing streptavidin surfaces. The presence of the longer 5000 Da chain increased the steric repulsion of the bimodal mushroom layer and thus decreased the net adhesive force when shorter chains were ligated. However, the 5000 Da chain did not detectably alter the distance where ligand-receptor binding occurs and adhesion begins. This latter result is in good agreement with theoretical predictions based on summing the repulsive steric and attractive bridging forces. Further, all ligated structures adhered to receptors under both static and dynamic fluid flow conditions. The dynamic movement of the flexible PEG tethers permitted ligand-receptor bonds to form far beyond the equilibrium edge of the bimodal mushroom layer. This work demonstrates that liposome targeting should be enhanced by grafting ligands to liposomes with a tether that has a contour length longer than the equilibrium height of the bimodal mushroom layer.     

In situ neutron reflectometry investigation of gold-chemisorbed PEO layers of varying chain density: relationship of layer structure to protein resistance

In work reported previously [L.D. Unsworth, H. Sheardown, J.L. Brash, Langmuir 21 (2005) 1036] we investigated protein interactions with polyethylene oxide (PEO) layers formed by chemisorption of thiol-PEO on gold. It was shown that, as a function of surface chain density, protein adsorption passed through a minimum. In follow-on work reported here, neutron reflectometry (NR) was used to investigate the formation and properties (volume fraction and chain density) of such PEO layers in situ, i.e., in contact with water. Chain density was varied by varying solubility conditions (far from and near the cloud point) and chemisorption time. Neutron experiments were carried out using neutrons of de Broglie wavelength 2.37 A. Contrast matching techniques were used to improve sensitivity. Layers formed under high solubility conditions were found to have PEO volume fraction, layer thickness and chain density of 0.33, 28 A, and 0.56 chains/nm2, respectively, after 0.5 h chemisorption; and 0.31, 28.5 A, and 0.59 chains/nm2, respectively, after 11 h, suggesting that the layer is fully formed within 0.5 h. Both chain density and PEO volume fraction in the chemisorbed layers were significantly greater when the layers were formed under low solubility conditions. The PEO layers shown in our previous work to have maximum protein resistance were found to have a PEO volume fraction of approximately 40%. Moreover the limiting volume fraction in the PEO films formed under low solubility conditions was approximately 57%, a value similar to the solubility limit of PEO in aqueous solution, suggesting that local regions in the layers may be phase separated under these conditions. This may result in increased hydrophobicity and may explain why protein adsorption was found to increase on the layers of higher chain density.     

Conformation of Chains in Cores of Block Copolymer Micelles with Solubilized Homopolymer: a Monte Carlo Study

A system of compatible self‐avoiding polymer chains solubilized in spherical cores of block copolymer micelles was studied by lattice Monte Carlo simulations. The core is modeled as a spherical cavity on a simple cubic lattice, filled in partially by tethered (core‐forming) chains and partially by free (solubilized) chains. Molecular parameters (e.g., the ratio of the contour length of the model chains to the core radius) correspond to those in real micellar systems. The density (the fraction of lattice sites occupied) is 0.6 which corresponds to swollen micellar cores in real micellar systems. Simulations yield a constant segment density profile in the core. Both the tethered and solubilized chains acquire an ellipsoidal shape. The ellipsoids equivalent to both types of chains are more spherical than those in a melt and strongly oriented. The chains in the core show a Gaussian‐like behavior. Minor deviations from Gaussian behavior for tethered chains are due to surface effects.     

Characterization of thermo-reversible gels by means of sedimentation equilibria

This paper describes the deformation of gels in a centrifugal field leading to a continuous equilibrium. A gel is considered to be a binary mixture of cross-linked polymer and solvent and is assumed to remain isotropic during the deformation. From the equation for the osmotically effective pressure, called swelling pressure, the thermodynamic properties of a gel can be calculated. For highly swollen gels the expression of Svedberg and Pedersen is obtained. It is shown that the complete concentration dependence of the swelling pressure in the concentration range of the maximally swollen gel up to that at the cell bottom can be measured in a single equilibrium experiment. The homogeneity of weakly cross-linked gels can be examined by means of the method described. Soluble parts which are not incorporated into the polymer network can also be detected if they are present. From the swelling pressure-concentration curves it is possible to derive the thermodynamic properties of the physically crosslinked gelatin/water gels that were investigated. These gels can be described by means of a slightly modified Flory-Huggins equation with an interaction parameter χw in the weight fraction scale, which depends linearly on concentration. The interaction parameters show a dependence on concentration which is explained by an increased branching and crosslinking of the polymer with increasing initial polymer concentration of the gels. At low initial polymer concentration, the primary chains have to aggregate to build up relatively long chains between the network junctions. The static shear modulus G which can be calculated from the network term has the same order of magnitude as the real part of the complex shear modulus which is measured at low frequency.     

Integral equation theory for athermal solutions of linear polymers

An integral equation model is developed for athermal solutions of flexible linear polymers with particular reference to good solvent conditions. Results from scaling theory are used in formulating form factors for describing the single chain structure, and the impact of solvent quality on the chain fractal dimension is accounted for. Calculations are performed within the stringlike implementation of the polymer reference interaction site model with blobs (as opposed to complete chains) treated as the constituent structural units for semidilute solutions. Results are presented for the second virial coefficient between polymer coils and the osmotic compressibility as functions of the chain length and polymer volume fraction, respectively. Findings from this model agree with results from scaling theory and experimental measurements, as well as with an earlier investigation in which self-avoiding chains were described using Gaussian form factors with a chain length and concentration-dependent effective statistical segment length. The volume fractions at the threshold for connectedness percolation are evaluated within a coarse-grained closure relation for the connectedness Ornstein-Zernike equation. Results from these calculations are consistent with the usual interpretation of the semidilute crossover concentration for model solutions of both ideal and swollen polymer coils.     

Simulation Study of the Conformational Properties of Diblock Polyelectrolytes in Salt Solutions

Coarse-grained molecular dynamics simulations are performed to understand the behavior of diblock polyelectrolytes in solutions of divalent salt by studying the conformations of chains over a wide range of salt concentrations. The polymer molecules are modeled as bead spring chains with different charged fractions and the counterions and salt ions are incorporated explicitly. Upon addition of a divalent salt, the salt cations replace the monovalent counterions, and the condensation of divalent salt cations onto the polyelectrolyte increases, and the chains favor to collapse. The condensation of ions changes with the salt concentration and depends on the charged fraction. Also, the degree of collapse at a given salt concentration changes with the increasing valency of the counterion due to the bridging effect. As a quantitative measure of the distribution of counterions around the polyelectrolyte chain, we study the radial distribution function between monomers on different polyelectrolytes and the counterions inside the counterion worm surrounding a polymer chain at different concentrations of the divalent salt. Our simulation results show a strong dependence of salt concentration on the conformational properties of diblock copolymers and indicate that it can tune the self-assembly behaviors of such charged polyelectrolyte block copolymers.     

Monte Carlo simulation of structure and nanoscale interactions in polymer nanocomposites

Off-lattice Monte Carlo simulations in the canonical ensemble are used to study polymer-particle interactions in nanocomposite materials. Specifically, nanoscale interactions between long polymer chains (N=550) and strongly adsorbing colloidal particles of comparable size to the polymer coils are quantified and their influence on nanocomposite structure and dynamics investigated. In this work, polymer-particle interactions are computed from the integrated force-distance curve on a pair of particles approaching each other in an isotropic polymer medium. Two distinct contributions to the polymer-particle interaction potential are identified: a damped oscillatory component that is due to chain density fluctuations and a steric repulsive component that arises from polymer confinement between the surfaces of approaching particles. Significantly, in systems where particles are in a dense polymer melt, the latter effect is found to be much stronger than the attractive polymer bridging effect. The polymer-particle interaction potential and the van der Waals potential between particles determine the equilibrium particle structure. Under thermodynamic equilibrium, particle aggregation is observed and there exists a fully developed polymer-particle network at a particle volume fraction of 11.3%. Near-surface polymer chain configurations deduced from our simulations are in good agreement with results from previous simulation studies.     

Computer simulation of solutions of telechelic polymers with associating end-groups

We present the results of numerical Monte Carlo simulations of solutions of telechelic chains with strongly attracting end-groups. Formation of micelles (aggregates), their structure and structural characteristics of the system as a whole are studied in detail. The features revealed in computer experiments are qualitatively similar to the recent theoretical predictions. In particular, we show that micelles formed by telechelic chains attract each other even if the solvent is good for the soluble blocks forming micellar shells. As a result, A “micellar gel” structure with a number of chain “bridges” connecting micelles is formed. The bridging chains turn out to be significantly stretched. Furthermore, we observe a pronounced maximum in the wave-vector dependence of the static structure factor for the associating end-units which is a manifestation of a quasiperiodic pattern of alternating microdomains consisting of dense micellar cores and the swollen soluble chain blocks.     

Hydrogen bonded clusters of alcohol molecules in inert solvents: a chain statistics approach

A theory is proposed for modeling the autoassociation of alcohol molecules either in the neat liquid or in inert (non-complexing) solvents. This autoassociation by hydrogen bonds takes the form of clusters which can be predominantly open or cyclized chains, depending on the concentration. Chain conformation statistics are used for calculating the dipole moment of each type of chain. Cyclization equilibrium theory is employed for obtaining formulae for the fraction of all types of clusters. The total Kirkwood angular correlation factorg is calculated in function of the concentration. For this only two parameters are used; the first one is a common equilibrium constant for H bond dissociation; the second one is closely linked to the stiffness of the chains. A numerical illustration is presented for the case of normal hexanol in various aliphatic solvents, for which several dielectric data exist in the literature.     

Cononsolvency of poly-N-isopropyl acrylamide (PNIPAM): Microgels versus linear chains and macrogels

Poly-N-isopropyl acryl amide (PNIPAM) is swollen in both pure water and pure methanol but collapses in mixtures of these solvents. In this review, this cononsolvency of PNIPAM in water/methanol mixtures is discussed. Experimental studies of linear PNIPAM chains and macrogels are compared to microgels. Theoretical studies are presented based on molecular dynamics simulation and quantum mechanical calculations as well as semi-empirical models. The different explanations for the cononsolvency available in the literature are introduced. Experiments show that all PNIPAM species collapse and re-swell at comparable methanol fraction in the mixture. Cross-linker density of macrogels and microgels has only slight influence on cononsolvency, whereas chain length of linear chains has a significant influence. Microgels provide advantages to study cononsolvency by en'abling a broader experimental approach. Furthermore, multi-sensitive microgels can be prepared, which contain compartments sensitive to different stimuli.     

Adsorption of randomly annealed polyampholytes at the silica-water interface

We have investigated the adsorption of randomly annealed polyampholytes containing [2-(dimethylamino)ethyl methacrylate)] (DMAEMA), methacrylic acid (MAA), and [3-(2-methylpropionamido)propyl] trimethylammonium chloride (MAPTAC) with various molar compositions. The adsorption was performed from dilute aqueous solutions onto silicon substrates. The adsorbed layers were characterized by reflectivity techniques such as reflectometry, ellipsometry, and neutron specular reflection. As expected for annealed polyampholytes, the adsorption was found to depend strongly on the pH, with a maximum within the isoelectric domain of the polyampholyte. The monomer volume fraction profiles of the adsorbed layers were determined from neutron specular reflection measurements. In the isoelectric domain, the polyampholyte chains adopt a compact conformation, with a layer thickness of about 60 A. The polyampholyte layer is as dense as the adsorbed layer of fully charged polyelectrolyte but much thicker. Finally, we found that changing the ratio of neutral units along the polyampholyte chain in the isoelectric domain had no significant effect on the concentration profile of the adsorbed layer.     

Adsorption of random copolymers by a selective layer: Monte Carlo studies

We use scaling arguments and computer simulations to investigate the adsorption of symmetric AB-random copolymers (RC) from a diluted solution onto a selective ABA layer. Depending on the ratio between the layer thickness and the size of excess blobs, d/xi, three regimes of RC adsorption are predicted. For large values of the layer thickness RC adsorption can be understood as adsorption on two selective interfaces where sequences of RC chains form bridges. When the layer thickness is of the order of xi, excess blobs are trapped in the layer and localize the copolymer chain strongly. If the layer thickness is very small a weak adsorption scenario is predicted where large loops are formed outside the layer. Our simulations using the bond fluctuation model are in good agreement with the scaling predictions. We show that chain properties display non-monotonous behavior with respect to the layer thickness with optimal values for d approximately xi. In particular, we discuss simulation results for density profiles, statistics of bridges, loops and tails formed by the adsorbed chains, as well as for the adsorption order parameter and free energy.     

End-evaporation dynamics revisited

We present analytical results on the so-called end-evaporation kinetics in equilibrium polymeric systems following a temperature jump (T jump). A T jump prepares the system with a nonequilibrium length distribution, after which it relaxes back to its equilibrium state. Starting from a master equation, we develop a mean-field analytical theory based on a generating function approach, which allows explicit approximate expressions for the monomer and dimer concentrations to be derived in a discrete setting; the concentrations of the other chains as well as the average chain length were shown to be entirely expressible in terms of the monomer and dimer concentrations. We find that the calculated monomer and dimer concentrations as well as the average chain length are in good agreement with numerical simulation results and do not suffer from some of the defects of earlier continuum theories. Furthermore, the relaxation was shown to take place in three different stages. The first stage comprises the very fast relaxation of the monomers to almost their equilibrium concentration; the other polymer chains have hardly relaxed. During the second stage, which is highly nonlinear, a redistribution of material at practically constant monomer density takes place. Only in the final stage of the relaxation process the chain concentrations approach their true equilibrium values. In this stage there are only very small shifts in the concentrations of chains, which are governed by extremely slow "indirect" monomer-mediated processes.     

Model Bridging Effects of Asymmetrical Triblock Copolymers

Summary: We have performed Monte Carlo simulations to study the bridging of symmetrical or asymmetrical triblock copolymers confined between two similar or different solid surfaces based on a simple lattice model. The influence of the molecular structure, surface separation, adsorption energy, chain composition, and the chain concentration on the fractions of chains with bridge, loop and dangling configurations are reported in detail. The results show that the largest bridging fraction is given only when symmetrical triblock copolymers are confined between two parallel surfaces with the same adsorption energy. The bridge fraction is decreased so long as the asymmetry of the copolymers or the difference between the two surfaces is enhanced. It was found also that the bridging fraction increases as the adsorption energy increases. The bridging fraction Ω_bridge under different separations, L_z, can be expressed as in various situations. On the other hand, by introducing a symmetry index ν, the influence of molecular structure of copolymers on the bridges can be illustrated approximately by a relation when the two surfaces are similar and the adsorption energy is not too high. Combining the two expressions, data of the bridge fractions for copolymers of different symmetries confined between surfaces with different separations can be described with a single equation, which, in some occasion, can be used for prediction.

Influence of molecular structure on the bridging fraction for .