Polyelectrolyte chains in poor solvent. A variational description of necklace formation

We study the properties of polyelectrolyte chains under different solvent conditions, using a variational technique. The free energy and the conformational properties of a polyelectrolyte chain are studied by minimizing the free energy F_N, depending on N(N - 1)/2 trial probabilities that characterize the conformation of the chain. The Gaussian approximation is considered for a ring of length 2⁴ < N < 2⁸ and for an open chain of length 50 < N < 200 in poor- and theta-solvent conditions, including a Coulomb repulsion between the monomers. In theta-solvent conditions the blob size is measured and found in agreement with scaling theory, including charge depletion effects, expected for the case of an open chain. In poor-solvent conditions, a globule instability, driven by electrostatic repulsion, is observed. We notice also inhomogeneous behavior of the monomer-monomer correlation function, reminiscence of necklace formation in poor-solvent polyelectrolyte solutions. A global phase diagram in terms of solvent quality and inverse Bjerrum length is presented. Received 7 June 2001 and Received in final form 17 October 2001     

Elastic moduli of single-crystal C60

The matrix of elastic constants of the fcc phase of solid C₆₀ has been determined experimentally from measurements of the the velocity of 5 MHz ultrasound in single-crystal samples with different crystallographic orientations. The following values were obtained for the elastic moduli: C ₁₁=14.9±0.9 GPa, C ₁₂=8.8±1.0 GPa, and C ₄₄=6.6±0.18 GPa. The results are compared with theoretical estimates of the elastic moduli and data obtained in previous measurements of the elastic characteristics of solid C₆₀. Fiz. Tverd. Tela (St. Petersburg) 40, 173–175 (January 1998)     

Elastic moduli of solid helium at megabar pressures

Abstract

Elastic moduli and equation of state of highly compressed solid He are calculated up to the pressure 1.5 Mbar. While calculating the energy of the crystal, pair interatomic interactions and three-body interactions are taken into account. Three-body contribution is significant to a great extent both for the equation ofstate and for the elastic moduli and makes considerable softening of the equation of state. On the other hand, vibrational characteristics, such as the sound velocity and the Debye temperature, do not strongly depend on the inclusion of non-additive interactions.     

Elastic moduli of vortex lattices within nonlocal London model

Vortex lattice (VL) elastic response is analyzed within the nonlocal London model which holds for high-kappa clean superconductors. The squash modulus vanishes at the field H( square) where VL undergoes a square-to-rhombus transition. For H>H( square), where the square VL is stable, the rotation modulus turns zero at H = H(r), indicating VL instability to rotations. The shear modulus depends on the shear direction; the dependence is strong in the vicinity of H( square) where the square VL is soft with respect to the shear along [110]. The H dependences of the moduli are evaluated for LuNi(2)B(2)C.     

Elastic moduli inheritance and the weakest link in bulk metallic glasses

We show that a variety of bulk metallic glasses (BMGs) inherit their Young's modulus and shear modulus from the solvent components. This is attributed to preferential straining of locally solvent-rich configurations among tightly bonded atomic clusters, which constitute the weakest link in an amorphous structure. This aspect of inhomogeneous deformation, also revealed by our in situ neutron diffraction studies of an elastically deformed BMG, suggests a rubberlike viscoelastic behavior due to a hierarchy of atomic bonds in BMGs.     

Elastic moduli and poisson ratio for amorphous polymers and glasses

We propose that the product of the density by the square of the rms velocity of strain waves, which exhibits the features typical of elastic moduli, be referred to as effective (or characteristic) elastic modulus. The ratio of the bulk compression modulus to the effective elastic modulus is a single-valued function of the Poisson ratio not only for crystals and glasses, but also for amorphous organic polymers. The effective elastic modulus may be helpful in analysis of anharmonism of lattice vibrations of deformable bodies.     

Elastic moduli of random network models of amorphous Ge and Si

The bulk and shear moduli of periodic random network models of amorphous Ge and Si are computed. The Young's modulus is reduced to about 90% of the average in the crystalline form.     

Elastic waves in cubic crystals with positive or negative anisotropy of second-order elastic moduli

Elastic waves in cubic crystals are considered. A new classification of cubic crystals is proposed based on their elastic properties. All cubic crystals are shown to be divided into crystals with a positive or negative anisotropy of their second-order elastic moduli. The vibrational-branch spectra of crystals of these two types differ qualitatively in shape. The angular dependences of the polarization vectors are analyzed. The transverse component in quasi-longitudinal vibrations in cubic crystals is shown to be small and can be neglected. The longitudinal component in quasi-transverse modes is not small: its maximum value is 16.5% for Ge and reaches 27% for KCl.     

Cylindrical molecular brushes under poor solvent conditions: microscopic observation and scaling analysis

Axial contraction of cylindrical molecular brushes of polymethylmethacrylate was observed by static light scattering and scanning force microscopy. Single brush molecules were visualized as worm-like particles whose length was almost three times shorter than the contour length of the backbone. A somewhat larger length was measured by light scattering in a good solvent. A scaling approach has been used to analyze the driving forces for the axial contraction and the conformation of the molecular brushes.Received: 27 November 2003, Published online: 25 March 2004PACS: 36.20.-r Macromolecules and polymer molecules - 36.20.Ey Conformation (statistics and dynamics) - 61.16.Ch Scanning probe microscopy: scanning tunneling, atomic force, scanning optical, magnetic force, etc.S.S. Sheiko: Current address: Department of Chemistry, University of North Carolina, Chapel Hill, NC 27599-3290, USAS.A. Prokhorova: Current address: University of Freiburg, IMTEK - Institute for Microsystem Technology, Georges-Koehler-Allee 103,S.A. Prokhorova: 79110 Freiburg, GermanyM. Möller: Present adress: Institut für Technische Chemie und Makromolekulare Chemie, Rhein-Westfälische Technische Hochschule Aachen, Veltmanplatz 8, 52062 Aachen, Germany     

Elastic moduli approximation of higher symmetry for the acoustical properties of an anisotropic material

The issue of how to define and determine an optimal acoustical fit to a set of anisotropic elastic constants is addressed. The optimal moduli are defined as those which minimize the mean-squared difference in the acoustical tensors between the given moduli and all possible moduli of a chosen higher material symmetry. The solution is shown to be identical to minimizing a Euclidean distance function, or equivalently, projecting the tensor of elastic stiffness onto the appropriate symmetry. This has implications for how to best select anisotropic constants to acoustically model complex materials.     

Elastic moduli of hard c‐Zr3N4 from laser ultrasonic measurements

The bulk and shear moduli of dense polycrystalline oxygen‐bearing c‐Zr₃N₄ were determined to be B₀ = 217(20) GPa and G₀ = 163(9) GPa, respectively, using laser ultrasonic technique combined with a numerical analysis of the sample porosity. While the obtained B₀ is in excellent agreement with the earlier high‐pressure compression measurements, the G₀ value is 70% higher than the previous estimate. Since both G₀ and hardness of the dense c‐Zr₃N₄ exceed those of γ‐Si₃N₄, c‐Zr₃N₄ vies for the rank of the third hardest material after diamond and cubic BN. Our results also support the suggestion that shear modulus is a robust predictor of hardness. (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Elastic moduli and internal friction of nanocrystalline Pd and PdSi as a function of temperature

Resonant ultrasound spectroscopy was used to study the elastic constants and internal friction of two nanocrystalline palladium samples over the temperature range 3–300 K. The first material, nc-Pd, had a grain size of 80–100 nm and a density 93% of that of single-crystal bulk palladium. The second material, nc-PdSi containing 0.5 at.% Si, had a grain size of 15–22 nm and a density 97% of the single-crystal value. The bulk and shear moduli were significantly reduced in the nc-Pd material from that expected based on single-crystal data, the effect being greater for the bulk modulus. The moduli of nc-PdSi were reduced 4–5% from that based on crystalline Pd. As compared to previous reports of the elastic moduli of nanocrystalline palladium (grain size 5–15 nm) the present values for the larger-grained nc-Pd are comparable, but the present values for the smaller-grained nc-PdSi are considerably higher. An internal friction peak and a modulus defect were found in the nc-Pd material, but not in the nc-PdSi material. These effects are attributed to a relaxation process at the grain boundaries. The temperature dependence of the moduli is similar to that of crystalline palladium and is strongly influenced by electronic effects.     

Elastic moduli of nc-TiN/a-Si3N4 nanocomposites: Compressible, yet superhard

Earlier measurements of elastic moduli of nc-TiN/a-Si₃N₄ nanocomposites of different composition and hardness by means of vibrating reed and surface Brillouing scattering, that yield Young’s and shear modulus, as well as the Poisson’s ratio, have been confirmed by high-pressure X-ray diffraction measurements, that yield bulk modulus. It is found that elastic moduli of all measured samples are essentially the same within relatively small error of measurements, and only slightly lower than that of pure TiN. The nanocomposites are superhard thanks to their unique nanostructure with strengthened SiN_x interface.     

Microheterogeneity in phenyl group modified inorganic/organic hybrid gels after aerosol drying or slow solvent evaporation

Sol-gel systems were prepared by co-hydrolysis and co-condensation of tetraethoxysilane (TEOS) and phenyltriethoxysilane (PhTES). The sols were transferred into silica gels by Evaporation Induced Self-Assembly (EISA) or Aerosol Assisted Self-Assembly (AASA) using a laboratory spray-dryer. The structural properties such as porosity and homogeneity/microheterogeneity of these different systems are compared by N(2) sorption measurements, thermal analysis (TG, DTG and DTA), (29)Si MAS NMR and (29)Si{(1)H} CP MAS NMR. The cross polarization of the AASA gels can be described with the conventional I-S dynamics of a homogeneous proton spin bath. The EISA gels are heterogeneous, and the I-I(*)-S model, or a bimodal I-S model, was employed for the simulation of CP dynamics. Microheterogeneities are observed by (1)H-(29)Si cross polarization on an EISA sample, whereas rapid drying (AASA) transfers the corresponding sol into homogeneous xerogels. The EISA gels are microporous after calcination at 923 K, and the AASA gels are dense.