intermediate frequencies, T u (q) interpolates smoothly between these two limiting behaviours 12 .The behaviour seen in Fig. 2 is consistent with KTB dynamics, if we identify the crossover with T KTB of an isolated bilayer. Above T KTB , the conductivity is predicted to scale according to 13±15 :The scaling function S(q/) is constrained by the physics of the high-and low-frequency limits. As q= !`, S must approach i/q in order for j to assume its superconducting form, equation (1). At low frequencies, S approaches a real constant S 1 (0) which characterizes the d.c. conductivity of the normal state. By comparing the measured complex conductivity to equation (2), we can extract both the phase stiffness and correlation time at each temperature. To analyse the experimental data in terms of equation (2), we note that the phase angle of the complex conductivity, J [ tan 2 1 j 2 =j 1 , equals the phase angle of S(q/). Therefore J depends only on the single parameter , and is independent of T 0 u . With the appropriate choice of (T), all the measured values of J should collapse to a single curve when plotted as a function of the normalized frequency q/. Knowing (T), T 0 u is obtained from a collapse of the normalized conductivity magnitude, (~=k B T 0 v jjqj=j Q , to jSq=j. Figure 3 shows the collapse of the data to the phase angle and magnitude of S. As anticipated, S approaches a real constant in the limit q= ! 0, and approaches i/q as q= !`.When analysed further, the data reveal a con®rmation of thermal generation of vortices in the normal state. In the KTB picture we expect that the d.c. conductivity will equal k B T/n f D© 2 0 , which is thè¯u x-¯ow' conductivity of n f free vortices with quantized¯ux © 0 , and diffusivity D (ref. 16). Together with equation (2), this implies that is a linear function of n f , that is, 0 n f a vc =£, where a vc is the area of a vortex core, £ [ T=T 0 u is the reduced temperature, and 0 [ p 2 S 1 0D=a vc . Moreover, we expect that n f will be a thermally activated function, except for T very close to T KTB . The activation energy is simply Ck B T 0 u , where C is a non-universal constant of order unity. It follows that the¯uctuation frequency depends exponentially on the reciprocal of the reduced temperature, 0 =£exp 2 2C=£. The inset to Fig. 3 is a plot of log(£) versus 1/£ which shows that the exponential relation is observed over nearly four orders of magnitude. This is direct evidence that vanishing of phase coherence in our samples re¯ects the dynamics of thermally generated vortices. From the slope and intercept of a straight-line ®t we obtain C 2:23 and 0 1:14 3 10 14 s 2 1 .In Fig. 4 we present the behaviour of the bare stiffness and phasecorrelation time obtained from our measurement and modelling of j(q). The main panel contrasts T 0 u with the dynamical stiffness T u (q) measured at 150 and 400 GHz. The inset shows t as a function of temperature together with hatching that highlights the region where t ,~=k B T.The parameters displayed in Fig. 4 suggest that while phase ...

High-resolution calorimetric studies have been made of the liquid crystal phase transitions for several dispersions of 70-Å-diam silica spheres ͑aerosil͒ in octylcyanobiphenyl ͑8CB͒ as a function of silica density S. The excess specific heat peaks associated with the nematic-isotropic (N-I) and the nematic-smectic-A (N-SmA) transitions both exhibit shifts to lower temperatures, decreases in the specific heat maximum values, and decreases in the transition enthalpies as S is increased. Two distinct regimes of S-dependent behaviors are observed with a crossover between them at S Х0.1 g cm Ϫ3. For lower silica densities, sharp second-order C p peaks are observed at the N-SmA transitions, characterized by effective critical exponents that decrease monotonically with S from the pure 8CB value toward the three-dimensional XY value, and two closely spaced but distinct first-order C p features are observed at the N-I transition. For higher silica densities, both the N-SmA and the N-I transitions exhibit a single rounded C p peak, shifting in temperature and decreasing in total enthalpy in a manner similar to that observed in 8CBϩaerogel systems. Small angle x-ray scattering data are qualitatively aerogel-like and yield temperature-independent mass-fractal dimensionalities for aerosil aggregates that differ for samples with silica densities above and below the crossover density.

Three different water−alcohol cosolvent systems were used to assemble mesoporous molecular sieve silicas with wormhole framework structures (previously denoted HMS silicas) from an electrically neutral amine surfactant (S°) and a silicon alkoxide precursor (I°). The fundamental particle size and associated textural (interparticle) porosity of the disordered structures were correlated with the solubility of the surfactant in the water−alcohol cosolvents used for the S°I° assembly process. Polar cosolvents containing relatively low volume fractions of C n H2 n +1OH alcohols (n = 1−3) gave heterogeneous surfactant emulsions that assembled intergrown aggregates of small primary particles with high textural pore volumes (designated HMS−HTx). Conversely, three-dimensional, monolithic particles with little or no textural porosity (designated HMS−LTx) were formed from homogeneous surfactant solutions in lower polarity cosolvents. Aluminum substituted Al-HMS−HTx analogues with high textural porosity and improved framework accessibility also were shown to be much more efficient catalysts than Al-HMS−LTx or monolithic forms of hexagonal Al-MCM-41 for the sterically demanding condensed phase alkylation of 2,4-di-tert-butylphenol with cinnamyl alcohol. Transmission electron microscopy (TEM) and small-angle X-ray scattering (SAXS) studies verified the textural differences between wormhole HMS and electrostatically assembled hexagonal MCM-41 and SBA-3 molecular sieves. Power law fits to the scattering data indicated a surface fractal (D s = 2.76) for HMS−HTx, consistent with rough surfaces. A second power law at lower-q indicated the formation of a mass fractal (D m = 1.83) consistent with branching of small fundamental particles. Hexagonal MCM-41 and SBA-3 silicas, on the other hand, exhibited scattering properties consistent with moderately rough surfaces (D s = 2.35 and 2.22, respectively) and large particle diameters (≫1 μm). HMS−LTx silicas showed little or no mass fractal character (D m = 2.87), and no surface fractal scattering.

High-resolution x-ray scattering studies of thin smectic-C (SmC) samples prepared between solid plates by cooling from the smectic-^ phase reveal a surprising "chevron" structure of tilted layers. This structure is formed as a response to the shrinking of the SmC layers while anchored to the solid plates. The layer tilt is independent of surface treatment. Our results provide fundamental new information on the structure and surface interactions of SmC layers and provide evidence for a new SmC defect.

Photopolymerizable diacrylate monomers dissolved in fluid-layer smectic A and smectic C liquid crystal (LC) hosts exhibited significant spatial segregation and orientation that depend strongly on monomer structure. Small, flexible monomers such as 1,6-hexanediol diacrylate (HDDA) oriented parallel to the smectic layers and intercalated, whereas rod-shaped mesogen-like monomers such as 1,4-di-(4-(6-acryloyloxyhexyloxy)benzoyloxy)-2-methylbenzene (C6M) oriented normal to the smectic layers and collected within them. Such spatial segregation caused by the smectic layering dramatically enhanced photopolymerization rates; for HDDA, termination rates were reduced, whereas for C6M, both the termination and propagation rates were increased. These polymerization precursor structures suggest novel materials-design paradigms for gel LCs and nanophase-separated polymer systems.

Carbon blacks (CB) demonstrate varied structural features on length scales from angstroms to micrometers. Widely used as fillers in polymers, carbon blacks improve the mechanical and electrical properties of the host material. In general, CB is a low-dimensional mass-fractal aggregate of carbonaceous primary particles. Here we investigate the effect of processing on the interpenetration of aggregates in CB/polymer composites by small-angle X-ray scattering (SAXS). We performed SAXS measurements on a series of N330/EPR and N330/HDPE composites containing different amounts of a commercially available N330 carbon black. N330/HDPE composites were prepared by two different methods. The first method is Brabender dispersion of depelletized N330 in molten polymer; the second method is high-shear mixing of depelletized N330 in HDPE dissolved in a good solvent and subsequent addition of this mixture to a poor solvent for polyethylene. SAXS experiments were performed at the University of New Mexico/Sandia National Laboratories SAXS Laboratory. Data were collected on the Bonse-Hart camera with a q-range of 0.003 < q < 1 nm -1 . This wide q-range probes the structure of both the primary carbon black particles and aggregates of these particles. Depelletized N330 displays two power law regimes from which we deduce a surface-fractal dimension Ds ) 2.3 for the primary particles and a mass fractal dimension Dm ) 1.8 for the aggregate. For pelletized N330, the mass fractal domain vanishes as a result of aggregate interpenetration and, therefore, loss of correlation between primary particles. For N330/HDPE composites prepared by the Brabender method, the SAXS curve is similar to that obtained for the depelletized samples, indicative of little to no interpenetration of the aggregates. For N330/HDPE composites prepared by the solvent method and N330/EPR composites the SAXS curve is similar to that obtained for pelletized N330, indicative of extensive interpenetration of the aggregates.

Using small‐angle x‐ray (SAXS), neutron (SANS), x‐ray diffraction and light scattering, we study the structure of colloidal silica and carbon on length scales from 4 Å < q−1 < 107 Å where q is the magnitude of the scattering vector. These materials consist of primary particles of the order of 100 Å, aggregated into micron‐sized aggregates that in turn are agglomerated into 100 µ agglomerates. The diffraction data show that the primary particles in precipitated silica are composed of highly defective amorphous silica with little intermediate‐range order (order on the scale of several bond distances). On the next level of morphology, primary particles arise by a complex nucleation process in which primordial nuclei briefly aggregate into rough particles that subsequently smooth out to become the seeds for the primaries. The primaries aggregate to strongly bonded clusters by a complex process involving kinetic growth, mechanical disintegration and restructuring. Finally, the small‐angle scattering (SAS) data lead us to postulate that the aggregates cluster into porous, rough‐surfaced, non‐mass‐fractal agglomerates that can be broken down to the more strongly bonded aggregates by application of shear. We find similar structure in pelletized carbon blacks. In this case we show a linear scaling relation between the primary and aggregate sizes. We attribute the scaling to mechanical processing that deforms the fractal aggregates down to the maximum size able to withstand the compaction stress. Finally, we rationalize the observed structure based on empirical optimization by filler suppliers and some recent theoretical ideas due to Witten, Rubenstein and Colby.

Small-angle x-ray scattering, nitrogen adsorption, and scanning tunneling microscopy show that a series of activated carbons host an extended fractal network of channels with dimension D p 2.8 3.0 (pore fractal), channel width 15 20 Å (lower end of scaling), network diameter 3000 3400 Å (upper end of scaling), and porosity of 0.3-0.6. We interpret the network as a stack of quasiplanar invasion percolation clusters, formed by oxidative removal of walls between closed voids of diameter of ϳ10 Å and held in registry by fibrils of the biological precursor, and point out unique applications. DOI: 10.1103/PhysRevLett.88.115502 PACS numbers: 61.10.Eq, 47.55.Mh, 61.43.Hv, 81.05.Rm Since the first experimental studies of fractal surfaces of disordered solids [1], it has been conjectured that situations may exist in which the pore space -as opposed to the solid, or the surface alone -is fractal. Such fractal networks of channels crisscrossing the solid, termed pore fractals or "negative image" of mass fractals, have attracted interest as a laboratory for unusual dynamics of confined processes, induced by the long-range correlation of the pore space. The scaling laws predicted relate the dynamic exponents to the fractal and spectral dimensions (spacefilling and branching properties) of the network and include anomalous diffusion, reaction, free motion [2], phase transitions [3], electric conduction of pore fluid [4], and hydrodynamic flow [5]. Here we report the first welldocumented case of a pore fractal.The network is a new member in the family of nanostructured carbons. Its channel width is 15 20 Å, comparable to the width of single-wall carbon nanotubes, but instead of forming an assembly of freestanding tubes or bundle of tubes, the channels are embedded in a solid and connected. The network is of multiple interest: (i) Its synthesis differs vastly from that of isolated nanotubes. Created by controlled oxidation, a mainstay of mass production of porous carbons, it promises to be a low-cost competitor of isolated nanotubes for gas storage. (ii) For gas storage, it has outstanding mechanical stability, nanofluidic properties (rapid transport through branched channels), and capacity (high porosity, condensation in high dimensions) compared to nanotube bundles [6]. (iii) It offers a stage for "chemistry in confined spaces" and control of pathways similar to zeolites and other microscopic vessels [7]. (iv) The extended scaling regime, created by what we believe is invasion percolation, provides a unique platform to compare predicted dynamic exponents with experiment. [8] have been put forth as pore fractals, but these proposals have been controversial or withdrawn [9]. Pore fractals are more challenging to ascertain than mass or surface fractals because they do not reveal their fractality when probed with material yardsticks: the pore-size distribution of a pore fractal is a delta function (mass and surface fractals give a power law), so an intruding nonwetting liquid, capillary condensate, or adsorbed layer will eithe...