Critical Percolation Concentration Research Articles

Crystallization and dynamical arrest of attractive hard spheres

Crystallization of hard spheres interacting with a square well potential was investigated by numerical simulations using so-called Brownian cluster dynamics. The phase diagram was determined over a broad range of volume fractions. The crystallization rate was studied as a function of the interaction strength expressed in terms of the second virial coefficient. For volume fractions below about 0.3 the rate was found to increase abruptly with increasing attraction at the binodal of the metastable liquid-liquid phase separation. The rate increased until a maximum was reached after which it decreased with a power law dependence on the second virial coefficient. Above a critical percolation concentration, a transient system spanning network of connected particles was formed. Crystals were formed initially as part of the network, but eventually crystallization led to the breakup of the network. The lifetime of the transient gels increased very rapidly over a small range of interaction energies. Weak attraction destabilized the so-called repulsive crystals formed in pure hard sphere systems and shifted the coexistence line to higher volume fractions. Stronger attraction led to the formation of a denser, so-called attractive, crystalline phase. Nucleation of attractive crystals in the repulsive crystalline phase was observed close to the transition.

Properties of a carbon filled cyclic olefin copolymer

AbstractThis article investigates electrical conductivity and rheological aspects of cyclic olefin copolymer (COC) composites containing both carbon fiber (CF) and carbon black (CB) at various concentrations. The different formulations of carbon filled COC were compression molded in such a manner that the formed circular sheets exhibited preferred in‐plane filler orientation. Through‐plane and in‐plane conductivity were measured by 2‐probe and 4‐probe methods, respectively, while an ARES rheometer in dynamic mode was employed to measure the storage modulus and complex viscosity. It was found that formulations with CF:CB ratios around 3 and where the CB content was close or below its critical percolation concentration resulted in higher electrical conductivity while maintaining the viscosity of the composite at a level acceptable for polymer processing machinery. For those composites containing both fillers, collaborative associations between the CB and CF fillers were found in the established percolating network structure, producing measured conductivities which exceeded the estimated values by the additive rule by up to sixfold. An empirical expression to handle hybrid filler systems is proposed in this work based on the standard percolation model. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 1808–1820, 2007

Dynamic percolation of carbon nanotubes in liquid medium

In this article we have measured the electrical behaviour of carbon nanotubes (CNTs) suspended in electrically insulating liquid. The concentration dependence of conductivity shows a percolation behaviour similar to that observed in electrical composites with an insulating matrix. The value of the critical percolation concentration is strongly determined by the aspect ratio of the fillers forming the network through dynamic percolation. We characterized several single- and multi-wall carbon nanotube materials by the newly proposed method and received a good correlation with the results obtained by methods commonly used for CNT characterization (Raman spectroscopy, transmission electron microscopy and electrical conductivity of free standing papers). As a comparison, fine graphite material has also been evaluated. The electrical properties of the suspensions can be used as a method for CNT characterization. This method can yield important information for CNT producers and for the selection of electrically conducting structures for composites applications.

Self-Assembly of Tin Oxide Nanoparticles: Localized Percolating Network Formation in Polymer Matrix

A novel method to prepare transparent antistatic films by trapping nanoparticles in thin surface layers is proposed, and high performance of the product is demonstrated. Coating solutions consisting of surface-active antimony-doped tin oxide (ATO) nanoparticles, an organic solvent mixture of high affinity with poly(methyl methacrylate) substrate, and an ultraviolet curable resin are used for the film formation. Antistatic property of the layer is obtained at ATO concentrations above the critical concentration for percolation. A scaling analysis of the data shows that the critical concentration (0.004 volume fraction) is extremely low as compared to the value predicted by the percolation theory for randomly packed systems as well as the values in ever developed composite films filled with ATO particles. Microscopic observation of the deposited layers indicates that percolating clusters of ATO particles are localized on the layer surfaces. The excellent electrical and optical properties of the layer are attributed to the characteristic microstructure.

An approach to one‐dimensional conductive polymer composites

AbstractA substantial approach to one‐dimensional (1D) electrically conductive composites was proposed which was based on the thermodynamic analysis of electric‐field‐induced particle alignment in a nonpolar thermoplastic polymer matrix. The process condition window was based on the real‐time exploration of dynamic percolation under different electric fields with carbon black (CB)‐filled polyethylene as a model. The CB content was the main factor of the process condition. Its upper limit was set as the critical percolation concentration at the thermodynamic equilibrium state without an electric field to eliminate the possibility of conductive network formation perpendicular to the electric‐field direction, whereas its lower limit the critical percolation concentration at the thermodynamic equilibrium state under a critical electric field (E*). A composite with CB content in this window, isothermally treated in an electric field not less than E*, showed conductivity in the electric‐field direction about 105 times larger than that in the perpendicular direction. A 1D cluster structure in the direction of the electric filed was confirmed with scanning electron microscopy morphology observations. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 184–189, 2005

Radiation resistant polymer–carbon nanotube nanocomposite thin films

We have studied degradation of poly(methyl methacrylate)–CNT nanocomposite thin films, using a UV-ozone and an e-beam radiation as a function of CNT concentration. We have shown that the addition of CNT fillers can have a dramatic reinforcement effect on the nature of degradation by both high-energy radiations, where polymer-free radicals are mainly responsible for the proliferation of degradation. In addition, CNT networks can effectively disperse the radiation. The saturation in the reinforcement effect was observed when a concentration of CNT was approximately 0.5 wt.%. This concentration was interpreted in terms of a critical concentration for percolation of the CNT network, and the result was consistent with the sheet resistivity measurement for which physical contact between CNT fibers was evident.

Disorder in diluted spin systems

The influence of substitutional disorder on the magnetic properties of diluted Heisenberg spin systems is studied with regard to the magnetic stability of ferromagnetic diluted semiconductors (DMS). The equation of motion for the magnon Green's function within Tyablikov approximation is solved numerically for finite systems. The resulting spectral density is then used to estimate the magnetization and Curie temperature of an infinite system. This method is suitable for any form of a ferromagnetic exchange interaction. Besides different lattices and spin magnitude $S$, exchange interactions of different range are examined. The results show that, for short-range interaction, no magnetic order exists below the critical percolation concentration, whereas a linear dependence of the Curie temperature on the concentration of spins is found for ferromagnetic long-range interaction.

Mesoscopic Properties of Semiflexible Amyloid Fibrils

We have determined the contour length, persistence length, bending rigidity, and critical percolation concentration for semiflexible amyloid fibrils formed from the globular proteins beta-lactoglobulin, bovine serum albumin, and ovalbumin. The persistence length was estimated using an adjusted random contact model for highly charged semiflexible chains. We have found contour lengths in the range of 50 nm to 10 microm and persistence lengths in the range of 16 nm to 1.6 microm. This wide range of contour and persistence lengths and the ease of preparation of these amyloid fibrils make them ideal model systems for the study of semiflexible polymers.

Dynamics of electric field induced particle alignment in nonpolar polymer matrix

The dynamics of electric field induced particle alignment in nonpolar polymer matrix to build one-dimensional conductive materials was investigated. The influence of electric field on particle alignment was real-time traced by dynamic percolation measurement using carbon black (CB) filled polyethylene as a model system. The activation energy of the continuous CB path formation was calculated and found to be unchanged with CB alignment. The critical percolation concentration at thermodynamic equilibrium state φc* was deduced to characterize the anisotropism of network structure, by which the thermodynamic prerequisite electric field E* for the transition from three-dimensional isotropic network to one-dimensional chain could be easily found out.

Irreversible self-assembly of ovalbumin into fibrils and the resulting network rheology

The self-assembly of ovalbumin into fibrils and resulting network properties were investigated at pH 2, as a function of ionic strength. Using transmission electron microscopy (TEM), the effect of ovalbumin concentration on the contour length was determined. The contour length was increasing with increasing ovalbumin concentration. TEM micrographs were made to investigate the effect of ionic strength on the contour length. In the measured ionic strength regime (0.01–0.035 M) fibrils of approximately equal length (±200 nm) were observed. TEM micrographs showed that the contour length of the fibrils, versus time after dilution, remained constant, which indicates that the self-assembly of ovalbumin is irreversible. Using the results of rheological measurements, we observed a decreasing critical percolation concentration with increasing ionic strength. We explain this result in terms of an adjusted random contact model for charged semiflexible fibrils. Hereby, this model has now been proven to be valid for fibril networks of β-lg, BSA and, currently, for ovalbumin.

Rheological Determination of Interaction Potential Energy for Aqueous Clay Suspensions

Swelling clay suspensions, considered as strongly anisotropic disks, are very unstable in high shear flows. In this study of sodium montmorillonites from Wyoming and China, a hard sphere model is shown to be relevant for the rheological modeling of dilute solutions (2−20 g/L). For more concentrated systems (20−80 g/L), orientation and alignment effects seem to be preponderant. High shear measurements give complete information on the average clay size and critical percolation concentration necessary for the yield stress appearance. At low shear, when interaction effects are dominant, an electrostatic double-layer approach gives directly the average interaction pair potential, allowing good predictions of the elasticity and yield stress of clay dispersions.

A new multistep Ca2+-induced cold gelation process for beta-lactoglobulin.

The objective of this study was to obtain beta-lactoglobulin (beta-lg) gels at very low protein concentrations using a new multistep Ca(2+)-induced cold gelation process. In the conventional cold gelation process, salt free beta-lg solutions were heated at neutral pH, cooled, and cross-linked by adding salts. In our new process, first, long linear beta-lg fibrils were formed at pH 2. Solutions of these fibrils were cooled, and subsequently, the pH was adjusted to 7 or 8. Transmission electron microscopy studies showed that the long linear fibrils formed at pH 2 were stable when the pH was adjusted to 7 or 8. In the final step, the fibrils were cross-linked using CaCl(2). Using rheological measurements, the critical percolation concentration was determined. In the new multistep cold gelation process, the critical percolation concentration was an order of magnitude lower than in the conventional cold gelation method.

Mesostructure of fibrillar bovine serum albumin gels

The mesostructure of bovine serum albumin (BSA) at low pH was investigated. Rheological measurements were performed to determine the critical percolation concentration ( c p). A decreasing c p with increasing ionic strength was found. Fibrils with a contour length of about 100–300 nm were found using transmission electron microscopy. The measured conversion of monomers into fibrils was independent of ionic strength (0.20–0.30 M). Dilution of BSA samples showed that the aggregation process is reversible and that there exists a critical concentration for the self-assembly of BSA. We explain the decreasing c p with increasing ionic strength in terms of an adjusted random contact model.

Effect of electrostatic interactions on the percolation concentration of fibrillar beta-lactoglobulin gels.

The effect of electrostatic interactions on the critical percolation concentration (c(p)) of fibrillar beta-lactoglobulin gels at pH 2 was investigated using rheological measurements, transmission electron microscopy (TEM), and performing conversion experiments. A decreasing c(p) with increasing ionic strength was found. The fraction of nonaggregated beta-lactoglobulin was independent of ionic strength in the regime of 0.01-0.08 M. TEM experiments showed long fibrils (2-7 microm) for ionic strengths between 0.01 and 0.08 M. Since both the conversion of monomers and the contour length of the fibrils were independent of ionic strength (0.01-0.08M), the linear increase of c(p) with the Debye length can be attributed purely to an increase of electrostatic repulsion between the fibrils. This increase is explained in terms of an adjusted random contact model which takes into account the charge and semiflexibility of the fibrils.

Analysis of charge hopping between redox center molecules in a polymer membrane based on percolation theory

Charge propagation in a polymer membrane dispersing redox center molecules was studied using Monte-Carlo simulation. The charge propagation was visualized using a uniform sphere sites model. The fraction of charge-transported sites (R CT) increased above concentration called percolation threshold. The charge transfer in a Nafion® membrane confining [(bpy)2 (H2O)RuORu(H2O)(bpy)2]4+ (abbreviated to Nafion [Ru-O-Ru]) as well as in a polymer pendant membrane was investigated with a potential-step chronoamperospectrometry (PSCAS), and the behavior was compared with the simulation The relationship between charge hopping distance and critical percolation concentration (c p) was obtained from the percolation theory and applied to the actual systems. The charge hopping distance between molecules was estimated as 2.04 nm for Nafion[Ru-O-Ru] system and 1.71 nm for polymer pendant . When the charge propagation takes place by charge hopping mechanism, a slow diffusion of the redox centers increases R CT at the concentration below c p, but decreases at the concentration above c p. Although the c p becomes ambiguous with increasing diffusion coefficient (D phys), it was suggested that c p value doesn't change. Copyright © 2001 John Wiley & Sons, Ltd.

Analysis of charge hopping between redox center molecules in a polymer membrane based on percolation theory

AbstractCharge propagation in a polymer membrane dispersing redox center molecules was studied using Monte‐Carlo simulation. The charge propagation was visualized using a uniform sphere sites model. The fraction of charge‐transported sites (R CT) increased above concentration called percolation threshold. The charge transfer in a Nafion® membrane confining [(bpy)2 (H2O)RuORu(H2O)(bpy)2]4+ (abbreviated to Nafion [Ru‐O‐Ru]) as well as in a polymer pendant $\hbox{Ru(bpy)}_{3}^{2+}$ membrane was investigated with a potential‐step chronoamperospectrometry (PSCAS), and the behavior was compared with the simulation The relationship between charge hopping distance and critical percolation concentration (c p) was obtained from the percolation theory and applied to the actual systems. The charge hopping distance between molecules was estimated as 2.04 nm for Nafion[Ru‐O‐Ru] system and 1.71 nm for polymer pendant $\hbox{\it Ru(bpy)}_{3}^{2+}$. When the charge propagation takes place by charge hopping mechanism, a slow diffusion of the redox centers increases R CT at the concentration below c p, but decreases at the concentration above c p. Although the c p becomes ambiguous with increasing diffusion coefficient (D phys), it was suggested that c p value doesn't change. Copyright © 2001 John Wiley & Sons, Ltd.

Behavior critical for bond diluted n-vector model in the effective field theory

In the present work the bond-diluted n-vector model on the simple cubic lattice is studied by applying the effective field theory for a cluster with two spins. The critical temperature and percolation concentrations for XY and Heisenberg models, as well the phase diagram in the T– p plane, respectively, are obtained and the results, when compared with the standard mean field approximation, are an improvement, even for small size clusters.

Study of universality class for site-bond correlated diluted spin systems in the n-vector model

Using the mean-field renormalization group approximation, the n-vector model is used to study the critical properties of the site-bond correlated diluted model on d-dimensional hypercubic lattices, by employing two different exchange interactions. The critical temperature and percolation concentrations are obtained. The phase diagram in the T– p plane on a simple cubic lattice for the classical Heisenberg ( n=3) model is presented. The values for the critical magnetic and thermal exponents for classical XY and the Heisenberg models, respectively, are obtained and the results, when compared with the other methods, are an improvement, even for small size clusters. In addition, the relation between the initial slopes for correlated and uncorrelated models is universal.

Finite Ferromagnetic Clusters La1−xZnxMnO3 (x = 0.05 and 0.10)

Detailed magnetic and nonlinear susceptibility measurements of the La1−xZn x MnO3 system are reported. For x = 0.05 and 0.1, a typical ferromagnetic (FM) signal in the field-cooled curve is observed at 38 K. However, for the insulating antiferromagnetic (AFM) parent compound LaMnO3 and for the x = 0.2 and 0.33 samples, the magnetization curves in the low temperature range, are smooth and flat. This observation confirms the coexistence of finite FM clusters in the insulating AFM region below the critical concentration (x c = 0.16) for percolation, predicted by the percolation model.

The transport mechanism in micro-crystalline silicon

We investigate the transport mechanism of μc-Si:H with a typical grain size in the 200 nm range. We study in particular a series of doped μc-Si:H samples with a wide range of crystallinities, x. The conductivity (and also the Hall mobility) vary by more than two orders of magnitude with crystallinity. In contrast, there is little variation of these quantities with temperature from room temperature to 10 K. We discuss these experimental data by comparison with theoretical network calculations of the dc conductivity and the Hall mobility for a random assembly of highly conducting crystalline and essentially non-conducting amorphous grains of concentrations x and (1− x), respectively. We show, that for x larger than the critical percolation concentration, p c, the sample conductivity can be less than the crystalline conductivity by more than two orders of magnitude. We further show that for x> p c, the Hall experiment, evaluated with the standard formulae valid for a homogeneous sample, yields the mean mobility of the mobile carriers to a good approximation. This agreement provides a theoretical basis for the use of Hall data in these inhomogeneous systems. The remaining temperature dependence of the conductivities for highly doped μc-Si:H is ascribed to the affects of barriers between the crystalline grains.