Unperturbed Radius Of Gyration Research Articles

Interphase Structure in Silica–Polystyrene Nanocomposites: A Coarse-Grained Molecular Dynamics Study

Silica nanoparticles (NPs) embedded in atactic polystyrene (PS) are simulated using coarse-grained (CG) potentials obtained via iterative Boltzmann inversion (IBI). The potentials are parametrized and validated on polystyrene of 2 kDa (i.e., chains containing 20 monomers). It is shown that the CG potentials are transferable between different systems. The structure of the polymer chains is strongly influenced by the NP. Layering, chain expansion, and preferential orientation of segments as well as of entire chains are found. The extent of the structural perturbation depends on the details of the system: bare NPs vs NPs grafted with PS chains, grafting density (0, 0.5, and 1 chains/nm2), length of the grafted chains (2 and 8 kDa), and the matrix chains (2–20 kDa). For example, there is a change in the swelling state for the grafted corona (8 kDa, 1 chains/nm2), when the matrix polymer is changed from 2 to > 8 kDa. This phenomenon, sometimes called “wet brush to dry brush transition”, is in good agreement with neutron scattering investigations. Another example is the behavior of the radius of gyration of free polymer chains close to the NP. Short chains expand compared to the bulk, whereas chains whose unperturbed radius of gyration is larger than that of the NP contract.

Polymer interphase structure near nanoscale inclusions: Comparison between random walk theory and experiment

We propose a theory for the structure of the interphase in polymeric nanocomposites based on the freely-jointed chain near an impermeable spherical inclusion. The theory considers only the entropic exclusion between a polymer and the inclusion. The radius of gyration of the polymer chain and its distortion from spherical symmetry are perturbed strongly when the chain end is closer than 3Rg0 from the surface of the sphere, where Rg0 is the unperturbed radius of gyration. Also, the maximum expansion of the chain is bounded by (4/3)½, equal to a 15% increase. The shape of the polymer chain is elongated tangentially to the surface when the chain end originates in the interphase, but elongates radially due to entropic repulsion when the chain end is very close to the surface of the sphere. All the distortions of the polymer conformation from ideality, and hence the spatial extent of the interphase, occur within the depletion region associated with colloids. The model is used to explain recent experimental results showing that polymer chain dimensions increase in the presence of nanoparticles. The model quantitatively reproduces the observations at low filler loadings using no adjustable parameters, suggesting that entropic arguments can describe the majority of the effect there.

Surface and interface viscoelastic behaviors of thin polymer films investigated by nanoindentation

Nanoindentation experiments were performed on poly(methyl methacrylate) films deposited on Si(100) substrates to investigate the effects of film thickness, loading rate, and molecular weight on the surface and interface viscoelastic behaviors. The film stiffness increased with the indentation depth due to the increase of the contact area and the effect of the rigid substrate. The film thickness and loading rate exhibited strong effects on the film stiffness, while the molecular weight effect was secondary. For film thickness greater than the unperturbed radius of gyration, the stiffness variation across the film thickness demonstrated the existence of three regimes of different viscoelastic behaviors. The loading rate effect on the viscoelastic behavior of the indented polymer films is interpreted in terms of the confinement between the indenter and the substrate surfaces that restricted the molecular chain mobility in the vicinity of the film/substrate interface. The through-thickness variation of the film stiffness (modified to exclude the viscous component) is examined in the context of a previous indentation model. The nanoscale viscoelastic behavior of the thin polymer films observed in this study has important implications in polymer nanostructuring and nanopatterning.

Conformational Study on Thin Films of Symmetric AnB2nAn Triblock Copolymer

AbstractSummary: The behavior of symmetric AnB2nAn triblock copolymer films confined between two hard neutral walls was explored by Monte Carlo simulation. The thicknesses of the films were between ≈1Rg0 and ≈7Rg0, where Rg0 is the unperturbed radius of gyration in the bulk. The confinement leads to a lamellar structure normal to the wall and the order‐disorder transition (ODT) temperature was found to be a function of film thickness. When the film thickness (D) was less than a critical value, DC, which is between 3Rg0 and 4Rg0, the ODT temperature (T*ODT) reduced by chain length N (T*ODT/N) decreased with decreasing film thickness. However, T*ODT/N was nearly independent of the film thickness when it was greater than DC. In the case of strong confinement (D < DC), the B block shrinks along the direction perpendicular to the wall and stretches along the direction parallel to the wall with decreasing film thickness, and the volume occupied by the B block shrinks. Under weak confinement conditions (D > DC), the volume of the B block is nearly independent of film thickness. The conformations of the B block in the disordered state are quite different from those in the lamellae. If the film is thick enough, the volume of the B block approaches its value in the unperturbed state, regardless of the morphology. When temperature decreases, the B block stretches in the direction perpendicular to the A/B interface and shrinks in the other two directions. In addition, decreasing the temperature leads to the chains adopting two main extreme conformations, coiling or stretching as much as they can. The scaling behavior of the fraction of bridge chains vs. the temperature obtained in the weak segregation limit was different from that predicted in the strong segregation limit.Schematic diagram of the X, Y and Z axis definition.magnified imageSchematic diagram of the X, Y and Z axis definition.

Theory of end-adsorbed polymer brushes in polymeric matrices

A detailed self-consistent field theory is used to calculate the properties of end-adsorbed polymer chains, or polymer ‘‘brushes,’’ in equilibrium with a blend of adsorbing B polymer and nonadsorbing A polymer. The brush properties depend on χab, Na/Nb and β, where χab is the Flory interaction parameter characterizing the thermodynamic interaction between the A and B polymers, Na and Nb are the respective degrees of polymerization of the A and B polymers and kBTβ is the net attractive interaction between the adsorbing chain end and the surface. For χab=0, β≫1 and Na/Nb≫1 an asymptotic limit is reached where the brush properties are completely specified by z*/Rbg, where z* is the integrated surface excess of the adsorbing polymer and Rbg is its unperturbed radius of gyration. In this strong adsorption, dry brush regime the volume fraction of adsorbing polymer at the surface increases rapidly from 0.0 to 0.96 as z*/Rbg increases from 0.0 to 1.6. For z*/Rbg>1.6 the surface volume fraction of the adsorbing polymer is close to one and the brush profiles are accurately described by a hyperbolic tangent form where the only adjustable parameter is a width parameter describing the interpenetration of the brush and the matrix polymer. This width decreases from 2Rbg at z*/Rbg=1.6 to 1.6Rbg at z*/Rbg=3.3. For high values of Na/Nb there is an attractive interaction between opposing polymer brushes, or between a single polymer brush and a nonadsorbing surface, which can be attributed to the free energy associated with the distortion of the matrix chains adjacent to the brush. Results are also extended to cases of weak adsorption characterized by low values of β.

Dependence of chain conformation and solution thermodynamics on composition for poly(ortho ester)s prepared from ketene acetals and mixtures of diols

AbstractThe solution properties of a series of random copolymers prepared from ketene acetals with mixtures of diols have been characterized by dilute solution light scattering, chromatographic, and viscometric techniques. Five solvents with refractive indices ranging from 1.4 to 1.5 were used to obtain the correct weight average molecular weight. Changes in the unperturbed radius of gyration, obtained by combined size exclusion chromatography and low‐angle light scattering techniques, parallelled that predicted by simple considerations of changes in repeat unit size. Thus the relative contributions of differing copolymer–solvent interactions and changing unperturbed radii of gyration to the hydrodynamic volume of random copolymers of different compositions were evaluated.

Interactions between mica surfaces in a polystyrene—cyclopentane solution near the θ-temperature

The forces F( D) between smooth mica surfaces immersed in cyclopentane have been measured as a function of their separation D both in the absence and in the presence of polystyrene (of two molar masses, 6 × 10 5 and 2 × 10 6) adsorbed onto the mica from the solvent, at a temperature (23 ± 1°C) slightly above the θ-temperature for this system (19.6°C). The interactions between the bare mica surfaces in the cyclopentane were short-ranged (≲10 nm) and attractive. After addition of polymer to the solvent the surfaces were incubated in the solution at small surface separation to limit the rate of polymer adsorption, and interactions were measured following progressively longer incubation periods. The following trends were noted for both molar masses: (i) Following initial adsorption of polymer, a long-ranged attraction between the surfaces was followed by an ultimate repulsion as the surfaces were compressed close together. (ii) At longer incubation times in the solution, the magnitude of the attraction decreased while the range of interaction and the ultimate repulsive regime moved to larger D values. (iii) After overnight incubation at large surface separation the interaction became monotonically repulsive, starting at a surface separation D ∼- 1.7 R g (unperturbed radius of gyration) for both molar masses used. This behavior did not change on further incubation. (iv) When the solution was replaced by pure solvent after overnight incubation, the monotonic repulsion changed to an initial weak attraction followed by an ultimate repulsion, very similar to the behavior following partial adsorption. Overall, the results indicate that attraction between the polymer covered mica surfaces is strongly correlated with extent of polymer adsorption, and disappears following full adsorption. The origin of the attraction is probably due to “bridging” effects, while the ultimate repulsion most likely occurs because of the dominance of osmotic interactions.

Distribution functions for Gaussian molecules. I. Stars and random regular nets

Distribution functions P (t) of the reduced squared radius of gyration are presented for linear, circular, and starlike polymer molecules represented in the unperturbed Gaussian approximation. Semiempirical functions of the form 〈s2〉0P(t) =C[1−tanh(a/tβ)]t (3ω1−2)/2exp[ −3λ1Tr(Λ0−1)t /2]where 〈s2〉0 is the mean square unperturbed radius of gyration and t=s2/〈s2〉0 fit the results of numerical quadratures quite well. Only a and β are adjustable parameters; Λ0 is the matrix of eigenvalues of the Kirchhoff matrix that describes molecular connectivity, while λ1 is its minimum eigenvalue with degeneracy ω1. Regular random nets consisting of μ junctions, each with the same functionality or valence f⩾3, have been generated by Monte Carlo methods; the eigenvalue spectra of their Kirchhoff matrices are described. The spectra are remarkably simple and regular in that they are linear functions. Random nets differ sharply in this regard from crystallographic lattices. The distribution functions of the radii of gyration of random networks are calculated to be nearly Gaussian in t; this is also demonstrated analytically. The dimensions of these molecules are extremely small.

Volume-Expansion Factor of ABA Block Copolymer

Some recent experiments show that the intrinsic viscosities of ABA-type triblock copolymers are sometimes smaller than those of the precursors B in the θ solvent for side blocks A.To explain this fact, a triblock copolymer is represented by three spherical segmental distributions connected with each other by four freely jointed sticks.The partition function is evaluated by the Bragg—Williams approximation with the lattice model. Expressions for the free energy and the volume-expansion factor of ABA-type triblock copolymers are obtained and conformational anomalies which have an effect upon the intrinsic viscosities are discussed by comparing the interaction energy and the unperturbed radius of gyration in the ring conformation with those for the averaged case.

Solution properties of high molecular weight polystyrene

AbstractA series of polystyrenes with weight‐average molecular weight M̄w up to 1.3 × 107 was prepared by anionic polymerization in tetrahydrofuran (THF). Each sample was characterized by gel‐permeation chromatography, light scattering, and viscometry. It was found that each sample had an almost symmetrical and very narrow molecular weight distribution (M̄w/M̄n < 1.07). The mean‐square unperturbed radius of gyration 〈S2〉0 was determined in trans‐decalin at 20.4°C as 〈S2〉0 = 7.86 × 10−18M̄w (cm2). The particle scattering factor was well represented by the Debye equation irrespective of solvent in the range of M̄w < 4 × 106, and only a small deviation was observed in benzene at higher molecular weights. The penetration function Ψ ≡ A2M2/4π3/2NA〈S〉23/2 was found to approach a relatively low asymptotic value of 0.21–0.23 at molecular weights above 2 × 106 in benzene at 30°C, where A2 is the second virial coefficient and NA is Avogrado's number. It was also found that the theta temperature in trans‐decalin was affected by the nature of polymer samples. A difference of about 3°C in the theta temperature was observed between two series of anionic polystyrenes, one prepared in THF and the other in benzene, but there was practically no difference in unperturbed chain dimension.

Polystyrenes of known structure: Part 3. Polymers with long-chain branching

The melt- and solution viscosity behaviour of some polystyrenes with long-chain branching is described. These polymers, which were prepared by reacting chloromethylated polystyrene with potassium polystyryl, are distinguishable from the comb-shaped polystyrenes previously described in having longer branches and lower branching frequencies. (The number-average molecular weight of the branches is greater than 4.6 × 10 4 and in some cases greater than that of the backbone.) Their low shear melt viscosities and intrinsic viscosities in tetrahydrofuran and cyclohexane are above those of the backbone polymer and increase as the branch length increases for a given branching frequency, the rate of increase being greater the lower the branching frequency. In each solvent the intrinsic viscosities of the branched polymers are below those of linear polymers of comparable molecular weights. The melt viscosities of the majority of the branched polymers are also below those of linear polymers of the same molecular weights, but for a few, those with the longest branches in the series with the lowest branching frequency, the opposite is true. Another outcome of this work is the finding that for these branched polymers the low-shear melt viscosities and the intrinsic viscosities in solution are given by: viscosity ∝〈S 2 0〉 a where 〈S 2 0〉 is the theoretical mean square unperturbed radius of gyration. The exponents a for the branched polymers in the melt and in tetrahydrofuran differ from those for linear polystyrenes for which a similar relation holds, but are about equal for branched and linear polystyrenes in cyclohexane. The behaviour shown by the branched polystyrenes described here is considered in relation to the behaviour of the comb-shaped polystyrenes reported earlier and that of the branched polystyrenes studied by Decker and by Fujimoto et al.