Power-law correlations and finite-size effects in silica particle aggregates studied by small-angle neutron scattering

Freltoft, T.; Kjems, Jørgen; Sinha, Sunil K.

doi:10.1103/physrevb.33.269

Cited by 457 publications

(300 citation statements)

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“…The power-law scattering behavior has been often related to two kinds of fractal dimensions; one is the mass fractal dimension d f and the other is the surface fractal dimension d s (Freltoft et al, 1986). Each fractal dimension is related to the scattering power-law behavior, I(q) / q À , by…”

Section: Hierarchy IIImentioning

confidence: 99%

Hierarchical structure of niobate nanosheets in aqueous solution

et al. 2007

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The hierarchical structure of an aqueous dispersion of niobate nanosheets was explored by using a combined method of ultra‐small‐angle and small‐angle scattering of neutrons and X‐rays. The concentration of the sheets studied was in the range where the dispersion exhibits a liquid‐crystal phase as evidenced by observation between crossed polarizers in a previous report. The scattering data covered a wide q scale of more than four orders of magnitude [3 × 10−4≤q≤ 10 nm−1, where q = (4π/λ)sin(θ/2), λ and θ being the wavelength of the incident beam and the scattering angle, respectively], corresponding to the length scale l = 2π/q from ~1 nm to ~20 µm. The scattering analyses provided information on the hierarchical structural elements including: (i) single nanosheets as a structure element (hierarchy I), (ii) parallel stacks of the sheets (hierarchy II), and (iii) spatial arrangements of the stacks (hierarchy III), in order of increasing length scale. Hierarchy II is closely related to the liquid‐crystal nature of the dispersion in which the spacing and the persistence length, normal and parallel to the stack surface, respectively, were disclosed. Hierarchy III gives rise to the low‐q upturn in the scattering profile, which may be characterized by mass‐fractal‐like power‐law scattering behavior. This finding is a surprise from the viewpoint of the liquid‐crystal nature of the dispersion, a possible model of which is proposed in the text.

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Section: Hierarchy IIImentioning

confidence: 99%

Hierarchical structure of niobate nanosheets in aqueous solution

et al. 2007

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“…Equation (2) represents the network as an "aggregate of spherical pores" (dimension D p , primary pore diameter L min ), with structure factor given by the term in brackets [16,18] and form factor e 2q 2 L 2 min ͞20 , using Babinet's principle. When networks touch, as they may, the overall network is fractal at scales below L max and uniform at scales above L max .…”

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confidence: 99%

Nearly Space-Filling Fractal Networks of Carbon Nanopores

et al. 2002

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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...

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“…20 The fractal nature of the system can be delineated from the power-law exponents derived from the scattering data. [21][22][23] If I(Q) varies as Q Ϫp in a Q region, where QR g ӷ1, then pϽ3 denotes that the scattering system may be a mass fractal 22,23 and 3ϽpϽ4 corresponds to a surface fractal. 21 The fractal dimension of the surface, D s , is D s ϭ6Ϫp.…”

Section: Resultsmentioning

confidence: 99%

Small-angle x-ray-scattering study of phase separation and crystallization in the bulk amorphousMg62Cu25
Liu
¹
,
Johnson
²
,
Schneider
³

et al. 1999
Phys. Rev. B

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We report on a small-angle x-ray-scattering ͑SAXS͒ and differential scanning calorimetry study of phase separation and crystallization in rapidly quenched amorphous Mg 62 Cu 25 Y 10 Li 3 alloy samples. Differential scanning calorimetry demonstrates the occurrence of crystallization and grain growth upon isothermal annealing of these samples at 135°C. The SAXS studies show the presence of large inhomogeneities even in the rapidly quenched as-prepared Mg 62 Cu 25 Y 10 Li 3 alloy that is attributed to phase separation in the undercooled liquid during the cooling process. After isothermal annealing at 135°C for longer than 30 min the samples exhibit a strong SAXS intensity that monotonically increases with increasing annealing time. During heat treatment, crystallization and growth of a nanocrystalline bcc-Mg 7 Li 3 phase occurs in the Y-poor and MgLirich domains. The initially rough boundaries of the nanocrystals become sharper with increasing annealing time. Anomalous small-angle x-ray-scattering investigations near the Cu K edge indicate that while Cu is distributed homogeneously in the as-prepared sample, a Cu composition gradient develops between the matrix and the bcc-Mg 7 Li 3 nanocrystals in the annealed sample. ͓S0163-1829͑99͒01018-8͔

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Power-law correlations and finite-size effects in silica particle aggregates studied by small-angle neutron scattering

Cited by 457 publications

References 11 publications

Hierarchical structure of niobate nanosheets in aqueous solution

Hierarchical structure of niobate nanosheets in aqueous solution

Nearly Space-Filling Fractal Networks of Carbon Nanopores

Small-angle x-ray-scattering study of phase separation and crystallization in the bulk amorphousMg62Cu25
Liu
¹
,
Johnson
²
,
Schneider
³

et al. 1999
Phys. Rev. B

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Power-law correlations and finite-size effects in silica particle aggregates studied by small-angle neutron scattering

Cited by 457 publications

References 11 publications

Hierarchical structure of niobate nanosheets in aqueous solution

Hierarchical structure of niobate nanosheets in aqueous solution

Nearly Space-Filling Fractal Networks of Carbon Nanopores

Small-angle x-ray-scattering study of phase separation and crystallization in the bulk amorphousMg62Cu25Liu1, Johnson2, Schneider3 et al. 1999Phys. Rev. B

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Small-angle x-ray-scattering study of phase separation and crystallization in the bulk amorphousMg62Cu25
Liu
¹
,
Johnson
²
,
Schneider
³

et al. 1999
Phys. Rev. B