Porod's Law Research Articles

Scaling for domain growth in the Ising model with competing dynamics

We study the domain growth of the one-dimensional kinetic Ising model under the competing influence of Glauber dynamics at temperature T and Kawasaki dynamics with a configuration-independent rate. The scaling of the structure factor is shown to have the form for nonconserved dynamics with the corrections arising from the spin-exchange process, i.e., $S(k,t)=Lg_0(kL,t/\tau )+g_1(kL,t/\tau)+... $, and the corresponding scaling functions are calculated analytically. A correction to the Porod law at zero temperature is also given.

Universality and scaling for the structure factor in dynamic order-disorder transitions

The universal form for the average scattering intensity from systems undergoing order-disorder transitions is found by numerical integration of the Langevin dynamics. The result is nearly identical for simulations involving two different forms of the local contribution to the free energy, supporting the idea that the model A dynamical universality class includes a wide range of local free-energy forms. An absolute comparison with no adjustable parameters is made to the forms predicted by theories of Kawasaki-Yalabik-Gunton, Ohta-Jasnow-Kawasaki, and Mazenko. The numerical results are well described by the Ohta-Jasnow-Kawasaki theory, except in the crossover region between scattering dominated by domain geometry and scattering determined by Porod's law.

Intrasolid Interfacial Fractality of a Less-Crystalline Solid

Small angle X-ray scattering (SAXS) measurements have been carried out on pitch (PIT) and cellulose (CEL)-based activated carbon fibers (ACFs). In the higher angle region, the scattering intensity did not obey the classical Porod's law. This suggests that ACFs have a rough surface and their roughness is expressed by the concept of surface fractal. The surface fractal dimension was determined from SAXS for each ACF. ACFs were treated at high temperature in argon in order to control the nanographitic crystallinity. PIT and CEL lost their microporosity upon heat treatment above 1773 and 2073 K, respectively. These nonporous ACFs showed also a strong SAXS caused by the electron density difference at the interface between microcrystalline and amorphous phase. This interface also had a fractal dimension, which was defined as the interfacial fractal dimension. The surface or interfacial fractal dimension of ACF depended on the heating temperature. As the treating temperature increased, the surface or interfacial fractal dimension decreased from 2.8 to 2.0. Both PIT and CEL showed a similar temperature dependence on each other. The surface or interfacial fractal dimension was reduced with the growth of nanographites, and upon heating at 3173 K, the intrasolid interfacial fractal dimension became 2.

The study of polystyrene blends: The relationship of phases and structure in blends of polystyrene with ethylene-propylene diene monomer rubber and its grafted copolymer

Pressed films of blends of polystyrene (PS) with ethylene–propylene diene monomer rubber (EPDM) or grafted copolymer of styrene (St) onto EPDM (EPDM-g-St) rubber were examined by small-angle X-ray scattering (SAXS), and scanning electron microscope (SEM). Small-angle X-ray scattering from the relation of phase was analyzed using Porod's Law and led to value of interface layer on blends. The thickness of interface layer (σb) had a maximum value at 50/50 (PS–EPDM-g-St) on blends. The radius of gyration of dispersed phase (domain) and correlation distances ac in blends of PS–EPDM-g-St were calculated using the data of SAXS. The morphology and structure of blends were investigated by SEM. © 1998 John Wiley & Sons, Inc. J. Appl. Polym. Sci. 70: 805–810, 1998

Structure Evolution of Spherulitic Crystallization in Semicrystalline Polymer Films

Isothermal nucleation-growth phenomena in 2D semicrystalline polymer films have been investigated by the time-resolved two-dimensional fast Fourier transform (2DFFT) as a quasi-scattering method via real-time in situ observation and measurements. The dynamic evolution of the structure function has been used to interpret the observed scattering intensity connected with crystallization kinetics. The important dynamic quantities, such as the intradomain scattering (domain-size distribution) and spatial distribution of nuclei, have been distinguished by the analysis of the power spectra for different crystallization conditions. The interdomain interference (spatial correlations) was interpreted by the aid of a pair correlation function. Experimental data confirm Porod's law, that is, the q-4 dependence of the tails. A scaling relation of the structure function is found experimentally for the free growth of domains during the postnucleation stage; however, it is not valid after impingement occurs.

Application of Small Angle Neutron Scattering to the in Situ Study of Protein Fouling on Ceramic Membranes

The fouling of ceramic ultrafiltration membranes by protein under dynamic filtration conditions has been investigated by small angle neutron scattering (SANS). Bovine serum albumin (BSA) solution was pumped through a thin commercial alumina membrane in a quartz cell specially designed to allow simultaneous SANS. SANS from clean membranes immersed in D2O obeyed Porod's law and showed that the pores in these ultrafiltration membranes more closely resemble fractal or random pore structures than well-oriented, monodisperse arrays of cylindrical pores. During protein filtration, the SANS patterns showed a buildup with time of protein on all parts of this internal surface. The effects of time and pH indicate that the performance with respect to filtration is closely linked to the buildup of the adsorbed layer of protein.

Morphology of Ionic Aggregates in Carboxylato- and Sulfonato-Telechelic Polyisoprenes As Investigated by Small-Angle X-ray Scattering

Twenty seven samples of carboxylato- and sulfonato-telechelic polyisoprenes associated with various cations (Na, K, Rb, Cs, Mg, Ca, Sr, and Ba) have been investigated by small-angle X-ray scattering. The Bragg spacing characteristic of the ionic peak is directly proportional to the root-mean-square end-to-end distance (rrms) of the polyisoprene chain. In the series of sulfonato-telechelic polyisoprenes, the Bragg spacing is approximately equal to rrms, whereas in the series of carboxylato-telechelic polyisoprenes it amounts to 21/2 rrms. It also appears that the ionic aggregates are more likely distributed according to a planar hexagonal network. An original method has been used for the tail-end analysis of the SAXS profile, which is based on the general vertex contribution to the Kirste−Porod law. It results that the ionic aggregates are of an angulous shape. Four different functions have been used to account for the interphase profile between the ionic phase and the polymeric matrix. The ionic aggregate...

Small angle X-ray scattering of aerogel densification

Silica aerogels were densified by isostatic compression or by thermal sintering. Small angle X-ray scattering experiments show that both types of aerogels exhibit oscillations around Porod behavior at high q values. Oscillations were modified by sintering but remain unchanged by compression. According to the Babinet's principle, it is not usually possible to associate intensity change with a peculiar phase of the porous solid. However, the fact that the oscillations remain unchanged with compression, which only acts on the porous part, indicates that SAXS intensity experiments can be directly linked to the solid part. Using the Porod law [G. Porod, Kolloid Z. 124 (1951) 83; P. Debye, A.M. Bueche, J. Appl. Phys. 20 (1949) 518], the specific surface area has been estimated. Surface area of the compressed aerogels does not change with density. In contrast, the specific surface area of the sintered samples decreases markedly. The absence of shape changes of SAXS curves indicates that the mean size of the solid phase remains constant during the compression densification.

Small-Angle X-ray Scattering Study of the Interfacial Characteristics Between δ' Phase and Matrix in Al–2.70 mass% Li Alloy

Small-angle X-ray scattering (SAXS) has been used to study the δ′ precipitation in an Al-2.70 mass% Li alloy aged at 463 K for 8 and 30 h. It is shown that the SAXS profile does not agree with Porod's law and has a negative slope. This suggests that the dispersive interfacial layer exists between the δ′ phase and the matrix. It is positive evidence that spinodal decomposition takes place during the precipitation process of the δ′ phase. The thicknesses of the interfacial layers are found to be 4.75 and 6.63 nm for the samples aged at 463 K for 8 and 30 h, respectively.

A Novel Approach to Extract Morphological Variables in Crystalline Polymers from Time-Resolved Synchrotron SAXS Data

A novel approach to extract morphological variables in crystalline polymers from time-resolved SAXS data using the method of correlation and interface distribution functions has been devised. The principle of the calculation is based on two alternative expressions of Porod's law using the form of the interference function. The approach enables the continuous estimate of the Porod constant, and corrections for liquid scattering and finite interface between the two phases, from the time-resolved data. A model of lamellar morphology has been implemented to interpret the calculated correlation and interface distribution functions. Many detailed morphological variables such as lamellar long period, thicknesses of crystal and amorphous phases, interface thickness, and scattering invariant can be estimated. An example analysis of isothermal crystallization in PET measured by synchrotron SAXS is demonstrated.

Small-Angle X-ray Scattering from Blends of PE and SBS. Observation of a Novel Kind of Deviation from Porod's Law

In a study of technical blends from poly(ethylene) and poly(styrene)–poly(butadiene) star block copolymers, reproducibility of the blending process and of the structural analysis is tested. A peculiar and reproducible deviation in the angular interval of the scattering curve is observed, which is governed by a decay according to Porod's law. The notion of a faint electron-density undulation in space with a wavelength of 5 nm, an amplitude of 1 e.u. nm−3 and a range of 40 nm could explain the triangular peak observed in a plot which otherwise linearizes the scattering curve in the Porod regime.

Late-stage coarsening dynamics in phase separating systems: simulation results

Simulations of diffusional interactions among precipitate particles with a given instantaneous size distribution and volume fraction are referred to as snapshot simulations. Dynamic (time-dependent) and snapshot simulations of large ensembles of coarsening particles with monopolar and dipolar interactions are presented. The snapshot simulations show that the deviation of coarsening rates from that of the infinite dilution limit of TLS varies as 3√V v or √V v (Debye screening) depending on the range of volume fraction and number of particles in the system. Dynamic simulation results of the same system confirm the cubic growth law of particles showing that they are in the asymptotic regime. The deviation of coarsening rates from TLS from dynamic simulations is larger than that predicted by the snapshot simulations at V v = 0.10 and higher. The results of the snapshot and dynamic simulations however agree at volume fractions ≤0.01. Structure factor calculations indicate that the scaled structure factor S(q) ~ q −4 (Porod's law) at large values of wave number q, consistent with the presence of sharp interfaces between the droplet and matrix phases.

SAXS measurements of the interface in polyacrylate and epoxy interpenetrating networks with fractal geometry

The interface thickness in two-component interpenetrating polymer networks (IPN) system based on polyacrylate and epoxy were determined using small-angle X-ray scattering (SAXS) in terms of the theory proposed by Ruland. The thickness was found to be nonexistent for the samples at various compositions and synthesized at variable conditions—temperature and initiator concentration. By viewing the system as a two-phase system with a sharp boundary, the roughness of the interface was described by fractal dimension, D, which slightly varies with composition and synthesis condition. Length scales in which surface fractals are proved to be correct exist for each sample and range from 0.02 to 0.4 Å−1. The interface in the present IPN system was treated as fractal, which reasonably explained the differences between Porod's law and experimental data, and gained an insight into the interaction between different segments on the interface. © 1997 Elsevier Science Ltd.

Blends of polypropylene with poly(cis-butadiene) rubber. I. Phase structure and morphology of blends

Pressed films of blends of polypropylene with poly(cis-butadiene) rubbers (PcBR) or grafted copolymer of ethyl acrylate (EA) onto PcBR (PcBR-g-EA) was examined by IR spectra, small angle X-ray scattering, small angle laser scatering (SALS), and scanning election microscopy (SEM). The problem of isomerization on PcBR (or PP/PcBR) is discussed by melt mixing (at a temperature of 180°C) using IR. X-ray scatering from the relation of phase was analyzed using Porod's law and led to the values of the interface layer of the blends. With increasing content of rubber, the interface layer thickness (σb) had a maximum value at 60/40 (PP/PcBR-g-EA) for the blends. The invariant (Q) was calculated and the relation of Q with σb was calculated using Debye-Buech statistical theory, and the average chord length (l) was obtained by Porod's definition of blends. The morphology and structure of the blends were investigated by SEM. The phase separation of the blends was found by SALS. © 1997 John Wiley & Sons, Inc. J Appl Polym Sci 64: 2265–2272, 1997

Small angle neutron scattering by opals

Micro- and non-crystalline opals, chalcedony and flint show diffuse small angle neutron scattering (SANS). Precious opals give rise to two additional intensity maxima at very small scattering angles which are due to Bragg reflections from the closest packed non-crystalline silica spheres. A small angle texture diagram reveals that the closest packing is faulty. Synthetic non-crystalline opals yield much less intense small angle scattering due to lower contrast between silica spheres and interstitial cement or particles; in this case intensity maxima were not observed. The outer part of the scattering curves of opal-CT and microcrystalline quartz deviates from Porod's law. The specific surface of natural non-crystalline opals ranges from 0.006 to 0.018 nm–1. In microcrystalline opals, the specific surface is about 10 times larger than in non-crystalline opals.

Determination of the transition region thickness in polymer blends by the SAXS method

The width of domain boundaries is an important parameter for the characterization of microphase separation in polymers. This parameter is often obtained by measurements of small-angle X-ray scattering (SAXS) in the range of validity of Porod's law, according to which X-ray intensity scattered by an ideal two-phase system decreases with the scattering vector h as h-4. Here h is defined as h = (4π sin)/λ, where is half the scattering angle and λ the wave-length of the X-ray. If the interface boundary is diffuse, i.e. the electron density does not change stepwise between separated phases but changes continuously in certain transition regions with a finite thickness, the intensity of scattering decreases more rapidly than predicted by Porod's law. In the present paper we have used the SAXS method in order to determine the supermolecular structure parameters, including the transition layer thickness, of polypropylene/polyamide-6 (PP/PA) blends. The transition layer thickness has been obtained by two methods elaborated by Koberstein and co-workers [4] and by Ruland [1], respectively. Both these methods assume that changes of the electron density in the transition region can be described by a Gaussian function with a standard deviation σ. The parameter σ have been determined graphically from the appropriate plots. Then, the thickness of the phase boundary, E, was estimated as √12σ. The investigated PP/ PA blends are multiphase systems and the problem of determination of the boundary width is more complicated because the meaning of Porod's law must be considered with caution. This problem is discussed based on a wide range of investigated samples prepared over various pressure and crystallization temperature conditions.

SANS of ZrO 2·SiO 2 gels

Small angle neutron scattering (SANS) measurements have been performed to investigate the nanoscale structure of materials of the system xZrO 2·SiO 2 with x≤10 mol%, at different processing stages. The materials were prepared by sol–gel using the alkoxides method, in strong acidic conditions. Samples studied as xerogels were heat-treated at 120° C and 850° C; as wet gels at the gel point; and after 4 h aging at 60° C. The samples showed extended linear chains ca. 10 nm long at the gel point. The aged gel has a mass fractal structure with fractal dimension of ca. 1.8. The 120°C heat-treated xerogel shows homogeneous oxide regions with 16 nm average diameter and mass fractal nature. For the 850°C heat-treated xerogel the oxide regions average diameter reduced to about 13 nm and presented a sharp and smooth surface, obeying the Porod law.

Ordering fractal aggregates of liquid mixtures

AbstractMixtures of liquid paraffin and polydimethylsiloxane (PDMS) were studied by small angle X‐ray scattering. The scattering patterns exhibited a single peak in finite angles and deviated from Porod's law in larger angles. The fractal dimension of mixtures was determined by power‐law analysis. The whole scattering curves were fitted well by a scattering formula. The average size and correlation distance of the aggregates were calculated from a correlation function. The fractal dimension and the static parameters were found to decrease with increasing the temperature and PDMS concentration.

SANS study of a fluorinated water-in-oil microemulsion

The experimental results of a SANS analysis of fluorinated water-in-oil microemulsions with perfluoropolyether oil and surfactant are shown at T = 20°C and T = 35°C for samples with a constant water-to-surfactant molar ratio W S = 11 . Under the hypothesis that the system is composed of interacting droplets, by a Guinier analysis of the dilute samples, a droplet radius of 23 Å is found for the two temperatures studied. Furthermore, all the samples studied follow the Porod law, as expected for dispersed particles with a sharp interface. For the two temperatures studied, a microemulsion interfacial area per surfactant molecule of 50 Å 2 is found in the dilute region and about 35 Å 2 in the concentrated region.

Porosity evolution during sintering in tungsten powders of different grain size

In recent work the densification behavior during sintering was investigated by neutron small angle scattering techniques. This experimental method allows the determination of the total porosity surface even in stages where the porosity is closed. In the literature most investigations of the sintering process deal with the time evolution of density at different temperatures. The time evolution of sintering is determined by various mechanisms like surface diffusion, grain boundary diffusion and bulk diffusion contributing to a varying extent to the densification during the different stages of sintering. Generally, the density change with time follows quite different curves if investigated as a function of temperature. Neutron small angle scattering measures the total porosity surface F via the Porod law which relates the scattered intensity I(Q) to F by the formula I(Q) {alpha} F/Q{sup 4}, where Q is the transfer of momentum. In the previous work on the sintering behavior of Mo, Nb, Ta and Al{sub 2}O{sub 3} ceramics the measured F values were related to the total cavity volume V during the different stages of the sintering process. The present work was undertaken in order to further corroborate the trajectory concept and, in particular, to investigate its dependence on the initialmore » state of the powder. The authors present a neutron small angle scattering investigation of the sintering behavior of a series of tungsten powders with several different initial grain sizes.« less