Bicontinuous geometries and molecular self-assembly: comparison of local curvature and global packing variations in genus-three cubic, tetragonal and rhombohedral surfaces

Balanced infinite periodic minimal surface families that contain the cubic Gyroid (G), Diamond (D) and Primitive (P) surfaces are studied in terms of their global packing and local curvature properties. These properties are central to understanding the formation of mesophases in amphiphile and copolymer molecular systems. The surfaces investigated are the tetragonal, rhombohedral and hexagonal tD, tP, tG, rG, rPD and H surfaces. These non-cubic minimal surfaces furnish topology-preserving transformation pathways between the three cubic surfaces. We introduce `packing (or global) homogeneity', defined as the standard deviation Δd of the distribution of the channel diameter throughout the labyrinth, where the channel diameter d is determined from the medial surface skeleton centered within the labyrinthine domains. Curvature homogeneity is defined similarly as the standard deviation ΔK of the distribution of Gaussian curvature. All data are presented for distinct length normalisations: constant surface-to-volume ratio, constant average Gaussian curvature and constant average channel diameter. We provide first and second moments of the distribution of channel diameter for all members of these surfaces complementing curvature data from [A. Fogden, S. Hyde, Eur. Phys. J. B 7, 91 (1999)]. The cubic G and D surfaces are deep local minima of Δd along the surface families (with G more homogeneous than D), whereas the cubic P surface is an inflection point of Δd with adjacent, more homogeneous surface members. Both curvature and packing homogeneity favour the tetragonal route between G and D (via tG and tD surfaces) in preference to the rhombohedral route (via rG and rPD).     

From 2D hyperbolic forests to 3D Euclidean entangled thickets

A method is developed to construct and analyse a wide class of graphs embedded in Euclidean 3D space, including multiply-connected and entangled examples. The graphs are derived via embeddings of infinite families of trees (forests) in the hyperbolic plane, and subsequent folding into triply periodic minimal surfaces, including the P, D, gyroid and H surfaces. Some of these graphs are natural generalisations of bicontinuous topologies to bi-, tri-, quadra- and octa-continuous forms. Interwoven layer graphs and periodic sets of finite clusters also emerge from the algorithm. Many of the graphs are chiral. The generated graphs are compared with some organo-metallic molecular crystals with multiple frameworks and molecular mesophases found in copolymer melts. Received 10 December 1999     

Rheological behaviour and shear thickening exhibited by aqueous CTAB micellar solutions

In this experimental work we carefully investigate the rheological behaviour and in particular the shear thickening exhibited by aqueous micellar solutions of CTAB with NaSal as added counterion. We are particularly interested in the evolution of the critical shear rate (at which shear thickening occurs) versus C _D , the surfactant concentration. We show that , at fixed salt concentration C _S , increases with C _D following a power law evolution with a positive exponent of + 5.8. On the other hand we show that if the ratio C _D / C _S is fixed, decreases with C _D with a negative exponent of -2.0. Nevertheless investigations of the zero shear viscosity indicate that in all situations (implying variation of the surfactant concentration C _D , or the salt concentration C _S or the temperature) is a decreasing function of the length of the micelles. All these evolutions are compatible with a gelation mechanism which could possibly be associated with entanglement effects of large interacting flowing structures. Received: 3 March 1998 / Revised: 16 June 1998 / Accepted: 3 July 1998     

Effects of finite thickness on interfacial widths in confined thin films of coexisting phases

The capillary broadening of a 2-phase interface is investigated both experimentally and theoretically. When a binary mixture in a thin film with thickness D segregates into two coexisting phases the interface between the two phases may form parallel to the substrate due to preferential surface attraction of one of the components. We show that the interfacial profile (of intrinsic width w₀) is broadened due to capillary waves, which lead to fluctuations, of correlation length of the local interface positions in the directions parallel to the confining walls. We postulate that acts as an upper cutoff for the spectrum of capillary waves on the interface, so that the effective mean square interfacial width w varies as . In the limit of large D this yields or respectively for the case of short- or long-range forces between walls and the interface. We used the Nuclear Reaction Analysis depth profiling technique, to investigate this broadening effect directly in two binary polymer mixtures. Our results reveal that the interfacial width indeed increases with film thickness D, though the observed interfacial width is lower than the predicted w. This is probably due to surface tension effects imposed by the confining surfaces which are not taken into account in our model. Received: 19 February 1998 / Received in final form: 2 September 1998 / Accepted: 8 September 1998     

The Stability of Several Triply Periodic Surfaces

The stability of six triply periodic surfaces of constant mean curvature (CMC) is investigated. The relative energy and curvature values of the surfaces comprising the P (Pm3¯m), I-WP (Im3¯m), and G (I4₁32) families are numerically calculated with K. Brakke's Surface Evolver. Regions where the I-WP surface can exist metastable to a complementary I-WP surface are found. This type of metastability is also found in the F-RD surface. Bifurcation points marking the stability limits of the P, I-WP, and G families are also calculated with Evolver. Modes of instability which may occur in the six CMC families are classified. Bifurcations in the P, G, I-WP, C(P), D, and F-RD families are attributed to fundamental instabilities. Lattices of spheres (LOS) are possible extremal surfaces at the bifurcations. It is determined that both the CMC surfaces and the LOS configurations are unstable to coarsening. Because the variation in curvature is lowest for the G family, it is the most robust of the six families to coarsening when the surfaces are otherwise equivalent.     

Topological representations of Jahn-Teller distortions

Topological representations (top-reps), which originally were developed to model molecular polyhedral isomerization processes, can be extended to depict the relationships between the polygons and polyhedra involved in Jahn-Teller (JT) distortions. Using this approach, the top-rep of the E ? (b _1g + b _2g) distortion of square planar molecules to rectangle and rhombus isomers becomes a rhombus in which the vertices alternately represent distortions to the rectangle and rhombus isomers. Similarly the top-rep of the E ? e distortion of regular O_h octahedra to elongated D_4h tetragonal bipyramids becomes a triangle in which the vertices represent the three distinct tetragonal bipyramids from a given octahedron and the edge midpoints represent lower symmetry rhombic D_2h intermediates. A regular octahedron can be used as a top-rep for the T ? (e + t ₂) distortions of regular octahedra if the 6 vertices represent distortions to D_4h tetragonal bipyramid isomers, the 8 face midpoints represent distortions to D_3d trigonal antiprism isomers, and the 12 edge midpoints represent lower symmetry rhombic D_2h intermediates. In the case of Jahn-Teller T ? h distortions of regular I_h icosahedra, the corresponding top-rep becomes a regular icosahedron in which the 12 vertices represent distortions to pentagonal D_5d isomers, the 20 face midpoints represent distortions to trigonal D_3d isomers, and the 30 edge midpoints represent D_2h intermediates. A 4-dimensional analogue of the tetrahedron (i.e. the 4-simplex) can be used as a top-rep for the G ? g problem and the H ? g component of the H ? (g + 2h) problem for JT distortions of regular icosahedra. In this case the 5 vertices and the 10 edge midpoints correspond to T_h isomers and D_3d intermediates, respectively.     

An universal relation between fractal and Euclidean (topological) dimensions of random systems

It is shown that a dimension-invariant form for fractal dimension D of random systems (where d is Euclidean dimension of the embedding space) is in good agreement with results of numerical simulations performed by different authors for critical (p=p _c ) and subcritical (p<p _c ) percolation, for lattice animals, and for different aggregation processes. Received: 9 July 1998 / Revised and Accepted: 12 July 1998     

Analysis of low temperature specific heat in the ferromagnetic state of the Ca-doped manganites

X-ray diffraction and Raman data on the pressure induced phase transitions of a nanometric zirconia, ZrO₂, are analyzed via a classical phenomenological Landau approach of the bulk. It is concluded that the initial tetragonal structure (D _4h ¹⁵), which is a metastable bulk state of zirconia at ambient conditions, evolves continuously towards the ideal cubic fluorite structure (O _h ⁵) via an intermediate tetragonal form (D _4h ¹⁴). The proposed phenomenological model describes consistently all experimental peculiarities, including the hybridization and softening of the low-frequency Raman active modes along with lattice-parameter anomalies.Received: 29 July 2003, Published online: 15 October 2003PACS: 64.70.Nd Structural transitions in nanoscale materials - 61.50.Ks Crystallographic aspects of phase transformations; pressure effects - 61.10.Nz X-ray diffraction     

On Bianchi and Bäcklund Transformations of Two-Dimensional Surfaces in E4

In this paper we discuss the existence and properties of the Bianchi transformations for pseudospherical surfaces in E ⁴. The results of the paper show that the theory of Bianchi transformations in the discussed case is essentially different from the well-known case of pseudospherical surfaces in E ³ (in general n-manifolds of constant and negative curvature in E ^{2 n − 1}). This revised version was published online in July 2006 with corrections to the Cover Date.     

The fluid-solid transition of Dzugutov's potential

We have studied fluid-solid phase transformations of materials interacting via the Dzugutov potential (Phys. Rev. A 46, R2984 (1992)). We present evidence from molecular dynamics simulations that this interaction does not exhibit a liquid phase. If a mixed potential (r) is formed by a linear superposition of and the Lennard-Jones potential , then the liquid phase disappears at a fraction of less than 60% . Received 15 June 1998 and Received in final form 8 July 1999     

Force distribution in an inhomogeneous sandpile

The force perturbation field in a two-dimensional pile of frictionless gravity-loaded discs or spheres arising from lattice distortions is derived to first order. The starting point is the model proposed by Liffman et al. (Powder Technology (1992) pp. 255-267) and Hong (Phys. Rev. E 47, 760-762 (1993)) in which discs of uniform size are arranged on a regular lattice: this predicts a uniform normal stress distribution at the base of the pile. The analysis is applied to two problems: (i) deformable (rather than rigid) grains that undergo Hertzian deformation at the points of contact; (ii) a pile containing a gradient in particle size from the centre to the free surfaces. The former results in the classical pressure dip at the centre; the latter also produces a dip if the larger particles are at the centre. Received 29 January 1998 and Received in final form 7 September 1998     

Dynamics of polymeric manifolds in melts: the Hartree approximation

The Martin-Siggia-Rose functional technique and the selfconsistent Hartree approximation is applied to the dynamics of a D-dimensional manifold in a melt of similar manifolds. The generalized Rouse equation is derived and its static and dynamic properties are studied. The static upper critical dimension, d _uc =2D/(2-D), discriminates between Gaussian (or screened) and non-Gaussian regimes, whereas its dynamical counterpart, , discriminates between Rouse- and renormalized-Rouse behavior. The Rouse modes correlation function in a stretched exponential form and the dynamical exponents are calculated explicitly. The special case of linear chains D=1 shows agreement with Monte-Carlo simulations. Received: 22 May 1998 / Received in final form: 31 August 1998 / Accepted: 8 September 1998     

Rapid and efficient synthesis of colloidal gold nanoparticles by?arc discharge method

The microstructural (XRD and SEM) and dielectric behavior of Pb(Zr_0.54Ti_0.46)O₃ (PZT 54/46) ceramic system with donor (La, Nb and La+Nb) doping was studied. For all Nb-doped PZT samples, only one (tetragonal) phase was found, which confirms the compositional shifts near the morphotropic phase boundary. For La- and La+Nb-doped samples, there are two (rhombohedral and tetragonal) phases. Dielectric characteristic behavior (1/ε) for La- and La+Nb-doped PZT was associated with two-phase transitions: Ferro–Ferro at low temperature and Ferro–Para at Curie temperature. For Nb-doped samples, only one phase transition is observed, which indicates the presence of a single ferroelectric phase.     

Phase transitions in “small” systems

Traditionally, phase transitions are defined in the thermodynamic limit only. We discuss how phase transitions of first order (with phase separation and surface tension), continuous transitions and (multi)-critical points can be seen and classified for small systems. “Small” systems are systems where the linear dimension is of the characteristic range of the interaction between the particles; i.e. also astrophysical systems are “small” in this sense. Boltzmann defines the entropy as the logarithm of the area of the surface in the mechanical N-body phase space at total energy E. The topology of S(E,N) or more precisely, of the curvature determinant allows the classification of phase transitions without taking the thermodynamic limit. Micro-canonical thermo-statistics and phase transitions will be discussed here for a system coupled by short range forces in another situation where entropy is not extensive. The first calculation of the entire entropy surface S(E,N) for the diluted Potts model (ordinary (q=3)-Potts model plus vacancies) on a square lattice is shown. The regions in {E,N} where D>0 correspond to pure phases, ordered resp. disordered, and D<0 represent transitions of first order with phase separation and “surface tension”. These regions are bordered by a line with D=0. A line of continuous transitions starts at the critical point of the ordinary (q=3)-Potts model and runs down to a branching point P_m. Along this line vanishes in the direction of the eigenvector of D with the largest eigen-value . It characterizes a maximum of the largest eigenvalue . This corresponds to a critical line where the transition is continuous and the surface tension disappears. Here the neighboring phases are indistinguishable. The region where two or more lines with D=0 cross is the region of the (multi)-critical point. The micro-canonical ensemble allows to put these phenomena entirely on the level of mechanics. Received 18 October 1999 and received in final form 17 November 1999     

A geometric generalization of field theory to manifolds of arbitrary dimension

We introduce a generalization of the O(N) field theory to N-colored membranes of arbitrary inner dimension D. The O(N) model is obtained for , while leads to self-avoiding tethered membranes (as the O(N) model reduces to self-avoiding polymers). The model is studied perturbatively by a 1-loop renormalization group analysis, and exactly as .Freedom to choose the expansion point D, leads to precise estimates of critical exponents of the O(N) model. Insights gained from this generalization include a conjecture on the nature of droplets dominating the 3d-Ising model at criticality; and the fixed point governing the random bond Ising model. Received: 15 October 1998 / Accepted: 4 November 1998     

Rational Surfaces Associated with Affine Root Systems¶and Geometry of the Painlevé Equations

We present a geometric approach to the theory of Painlevé equations based on rational surfaces. Our starting point is a compact smooth rational surface X which has a unique anti-canonical divisor D of canonical type. We classify all such surfaces X. To each X, there corresponds a root subsystem of E ⁽¹⁾ ₈ inside the Picard lattice of X. We realize the action of the corresponding affine Weyl group as the Cremona action on a family of these surfaces. We show that the translation part of the affine Weyl group gives rise to discrete Painlevé equations, and that the above action constitutes their group of symmetries by B?cklund transformations. The six Painlevé differential equations appear as degenerate cases of this construction. In the latter context, X is Okamoto's space of initial conditions and D is the pole divisor of the symplectic form defining the Hamiltonian structure. Received: 18 September 1999 / Accepted: 29 January 2001     

High-pressure synthesis, structural and Raman studies of a two-dimensional polymer crystal of

Two-dimensional polymerisation of a C₆₀ single crystal has been obtained under high-pressure high temperature conditions (700 K - 2 GPa). Crystalline order is preserved but the crystal splits into variants (orientational domains). The analysis of X-ray diffraction and Raman spectroscopy data reveals that the polymer crystal is primarily tetragonal with some admixture of rhombohedral phase. Furthermore, Raman spectroscopy gives evidence for additional C₆₀-C₆₀ dimers, which are probably disordered. For the tetragonal phase, it is shown that successive polymer layers are rotated by about the stacking axis, according to the P4₂/mmc space group symmetry. The structure of the rhombohedral phase is also clarified. The role of the interlayer interactions in stabilising the two-dimensional polymer phases of C₆₀ is discussed. Received 8 October 1999     

A geometrical model for the double octahedral group based on a genus two Riemann surface

Geometrical models of double point groups are provided by double-sheeted Riemann surfaces of nonzero genus. This avoids the geometrically confusing picture of the doubling operation R as a rotation by 2π (i.e. 360°), which should instead be the identity operation E. In this manner a Riemann surface of genus 2 doubly covering a sphere and platonically tessellated b 16 equilateral triangles provides a geometrical model for the double octahedral group ²O_h. Stretching this Riemann surface along one axis to convert the 16 equilateral triangles to isosceles triangles and the underlying sphere to a prolate ellipsoid provides a model for the ²D_4h double group arising from the Jahn-Teller elongation of the regular octahedron into a prolate tetragonal bipyramid.     

Broken supersymmetry in the matrix model on a circle

We consider the discretization of aD=2 surface using polygons. We map the surface onto superspace and integrate over surfaces of arbitrary genus, obtaining a discretized version of the Green-Schwarz string inD=1. Taking an unusual critical limit of the supersymmetric matrix model involved, we construct exact solutions, to all perturbative orders, for the discretized superstring in one dimension, both when the target space is a real line and when the theory is represented in terms of matrix variables on a circle of finite radius. We comment on the behavior of the compactfied perturbative expansion under duality transformations.BITNET: BELLUCCI at IRMLNF