Critical Percolation Concentration Research Articles

Heterogeneous Percolation in Poly(methylvinylsiloxane)/Silica Nanocomposites: The Role of Polymer–Particle Interaction

Agglomeration and linking of nanoparticles or aggregates lead to the formation of percolated particle networks and alter the mechanical enhancement in polymer nanocomposites. Although critical behaviors have been widely studied, the solidlike behavior is only well described under the condition of the uniform dispersion of nanoparticles. In this work, we illustrate the role of particle–polymer interaction in the uniformity of particle dispersion and mechanical enhancement. Two types of silica with different interfacial adhesion energies with poly(methylvinylsiloxane) were adopted, resulting in different uniformities of particle dispersion. The critical percolation concentrations from the yield shear stress and yield first normal stress difference are identical. A lower interfacial adhesion energy leads to a higher critical concentration. The preshear stress only affects the critical concentration but does not change the critical exponents, which rely on the particle–polymer interaction. The mechanical enhancement, expressed as the power-law dependence of the yield stresses on the filler content, exhibits extraordinarily large power-law exponents for nanocomposites with lower interfacial adhesion energy, seriously deviating from the theoretical prediction in homogeneous dispersion systems. Based on the structural information from small-angle X-ray scattering (SAXS)/ultrasmall-angle X-ray scattering (USAXS) and transmission electron microscopy (TEM), we propose a model describing the heterogeneous percolation of loose aggregates in the presence of compact aggregates. This model shows that the heterogeneity of aggregates, including the fraction of compact aggregates and their fractal dimension, is the key factor in the scaling relationship between the yield stress and the particle volume fraction.

Spin-wave theory in a randomly disordered lattice: A Heisenberg ferromagnet

Starting from the hamiltonian for the Heisenberg ferromagnet which comprise randomly distributed nonmagnetic ions as impurities in a Bravais lattice, we express the spin operators by means of the Dyson–Maleev transformation in terms of the Bose operators of the second quantization. Then by using methods of quantum statistical field theory, we derive the partition function and the free energy for the system. We adopt the Matsubara thermal perturbation method to a portion of the hamiltonian which describes the interaction between magnons and the stationary field of nonmagnetic ions. Upon averaging over all possible distributions of impurities, we express the free energy of the system as a function of the mean impurity concentration. Subsequently, we set up the double-time single particle Green function at temperature T in the momentum space in terms of magnon operators and derive the equation of motion for the Green function through the Heisenberg equation of motion and then solve the resulting equation. From this, we calculate the self-energy and then the spectral density function for the system. We apply the formalism to the case of the simple cubic lattice and compute the density of states, the spectral density function and the lifetime of the magnons as a function of energy for several values of the mean concentration of nonmagnetic ions in the ferromagnetic lattice. We calculate the magnon energy spectrum as a function of the average impurity concentration fraction c, which shows that for low lying states, the excitation energy increases continuously with c in the studied range 0.1≤c≤0.7. We also use the spectral density function to compute some thermodynamical quantities through the magnon occupation number. We have obtained closed form expressions for the configurationally averaged physical quantities of interest in a unified fashion as functions of the mean concentration of nonmagnetic impurities c to any order of c applicable below a critical percolation concentration cp. The quantities of interest comprise the thermodynamic potential (free energy), the spin-wave self-energy and the spectral density function from which other quantities can be derived.

Tumor Hypoxia Heterogeneity Affects Radiotherapy: Inverse-Percolation Shell-Model Monte Carlo Simulations.

Tumor hypoxia was discovered a century ago, and the interference of hypoxia with all radiotherapies is well known. Here, we demonstrate the potentially extreme effects of hypoxia heterogeneity on radiotherapy and combination radiochemotherapy. We observe that there is a decrease in hypoxia from tumor periphery to tumor center, due to oxygen diffusion, resulting in a gradient of radiative cell-kill probability, mathematically expressed as a probability gradient of occupied space removal. The radiotherapy-induced break-up of the tumor/TME network is modeled by the physics model of inverse percolation in a shell-like medium, using Monte Carlo simulations. The different shells now have different probabilities of space removal, spanning from higher probability in the periphery to lower probability in the center of the tumor. Mathematical results regarding the variability of the critical percolation concentration show an increase in the critical threshold with the applied increase in the probability of space removal. Such an observation will have an important medical implication: a much larger than expected radiation dose is needed for a tumor breakup enabling successful follow-up chemotherapy. Information on the TME’s hypoxia heterogeneity, as shown here with the numerical percolation model, may enable personalized precision radiation oncology therapy.

Nanosized carbon black as synergist in PP/POE-MA/IFR system for simultaneously improving thermal, electrical and mechanical properties

Nanosized carbon black (CB) was introduced into polypropylene/maleic anhydride-grafted polyolefin elastomer/intumescent flame retardant (PP/POE-MA/IFR) system to investigate the effect of nanofiller as synergist on thermal, electrical and mechanical properties of polymer composites. With 5 mass% CB into PP/POE-MA/IFR system (POFC5), the Tmax (corresponded to the temperature at the maximum mass loss rate) under air was increased by 122.4 °C; its limited oxygen index was as high as 31.4%; its vertical burning rating (UL-94) reached V0, and the peak value of heat release rate was decreased to only 19% of neat PP in cone calorimeter testing. Moreover, PP composites exhibited good electrical conductivity with more than 1.6 mass% CB, which is a low loading level to reach the critical percolation concentration. In addition, a good balance on stiffness and toughness of PP composites was achieved; especially, Young’s moduli and impact strength of POFC5 were increased to 1.26 and 2.5 times in comparison with that of neat PP, respectively. These results indicated that CB was an effective synergist in multi-component PP composites to simultaneously improve thermal, electrical and mechanical properties.

A reactive hydrophobically modified ethoxylated urethane (HEUR) associative polymer bearing benzophenone terminal groups: Synthesis, thickening and photo-initiating reactivity

Hydrophobically modified ethoxylated urethanes (HEURs), as a representative class of associative polymers, are widely used as rheology modifiers in waterborne systems such as coatings, emulsions, inks etc. due to their unique associative network structure and finite relaxation behavior in solution. However, conventional HEURs are non-reactive and generally stay free in final cured films, which may induce some damages to the performance of the coatings. To solve the problem, a new concept of reactive HEUR has been proposed. That is, a reactive HEUR should bear reactive hydrophobic end-groups which can chemically crosslink with matrix resins during the curing process of waterborne coatings. In this work, with the help of the hydrogen abstraction photo-initiating reactivity of benzophenone (BP) group, a reactive HEUR (BPC11HEUR) end-functionalized by a photo-initiator BP moiety has been prepared. In aqueous solution, as demonstrated by steady shear measurement, an exponential increment of solution viscosity along with the increasing of polymer concentration (C) was found when C surpassed its critical percolation concentration (Cp ≈ 23.6 g/L). Besides, ultraviolet (UV) light induced photolysis in aqueous solution manifested a typical photochemical reaction behavior of BPC11HEUR polymer with a final photolysis degree (D) of approximate 78% after 300 s of irradiation. In situ 1H NMR and rheological measurements collectively revealed the efficient photo-initiating reactivity of BPC11HEUR for waterborne UV monomers or resins. Further, rheological measurements demonstrated that the emulsion zero/low shear viscosity can be greatly enhanced by ca. 50 folds for a commercially available waterborne UV polyurethane acrylate emulsion when the addition amount of BPC11HEUR varied from 0 to 1.0 wt%. These results shown that the reactive BPC11HEUR polymer can not only be used to efficiently modify the rheological properties of solutions or emulsions but also can chemically crosslink with waterborne UV acrylic monomers or resins by light irradiation. The work may be of interest for development of reactive HEURs used in waterborne UV coatings, inks, and adhesives, etc.

Nanocomposites of LLDPE and Surface-Modified Cellulose Nanocrystals Prepared by Melt Processing.

Cellulose nanocrystals (CNCs) were surface modified by esterification in tetrahydrofuran (THF) at 25 °C using different catalysts and anhydrides bearing different alkyl side chain lengths. Unmodified and acetic anhydride (AcAnh)-modified CNCs were studied as potential nanofillers for linear low-density poly(ethylene) (LLDPE). Nanocomposites were prepared by melt processing. Determination of the size and size distribution of CNCs in the nanocomposites by SEM revealed an enhanced compatibility of the AcAnh-modified CNCs with the LLDPE matrix, since the average size of the aggregates of the modified CNCs (0.5–5 μm) was smaller compared to that of the unmodified CNCs (2–20 μm). Tensile test experiments revealed an increase in the nanocomposites’ stiffness and strain at break—by 20% and up to 90%, respectively—at the CNC concentration of 5 wt %, which is close to the critical percolation concentration. Since the CNC nanofiller simultaneously reduced LLDPE crystallinity, the reinforcement effect of CNCs was hampered. Therefore, the molding temperature was increased to 120 °C, and, in this way, the greatest increase of the Young’s modulus was achieved (by ~45%). Despite the enhanced compatibility of the AcAnh-modified CNCs with the LLDPE matrix, no additional effect on the mechanical properties of the nanocomposites was observed in comparison to the unmodified CNC.

Bacterial cellulose based flexible multifunctional nanocomposite sheets

Bacterial cellulose is an ideal material that is sustainable, biodegradable and inherently capable of functionalization. Hence this has been functionalized with a single component, Ni to exhibit multiple functionalities such as electrical conductivity, magnetic sensitivity as well as catalytic activity in both dried and hydrogel forms. A novel, simple ‘inverse chemical reduction’ technique has been developed to incorporate this component and make bacterial cellulose multifunctional. This technique mercerizes and opens the interfibrillar spaces which results in the formation of nanoparticles that lead to percolating paths for conduction. The flexible sheet becomes electrically conducting with just 20 vol% of nanoparticles in the composite as determined by thermogravimetry. The room temperature electrical conductance increases by about 7 orders of magnitude, 10−6–10 S on changing the Ni-precursor solution concentration from 0.015 to 0.02 M, indicating this to be the critical concentration for conduction percolation. The composites are highly magnetic at room temperature with a maximum energy product of 140 J m−3, comparable to some of the commercially available bonded oxide magnets. The hydrogel form of the nanocomposite is found to be catalytically active. The catalytic activity is retained even after leaving the nanocomposite hydrogel in water for 12 h in water.

Electrical-percolation effects in micellar solutions of alkyltrimethylammonium bromides

Specific conductivity K of aqueous solutions of alkyltrimethylammonium bromides has been studied in a wide range of concentrations c of surfactants containing 10, 12, 14, and 16 carbon atoms in alkyl chains. In general, three break points have been observed in the K(с) dependences. The first point observed upon increasing overall solution concentration corresponds to critical micelle concentration CMC1. The CMC1 values of alkyltrimethylammonium bromides decrease with an increase in the alkyl chain length. They are in satisfactory agreement with the published data. It has been supposed that the second break point in the K(с) dependences corresponds to the formation micellar structures as clusters and the appearance of channels with a higher specific conductivity, which is provided by the contribution from the overlap of electrical double layers existing in the vicinities of micelles. Surfactant concentrations corresponding to these break points have been called “critical percolation concentrations” (CPCs). The position of a CPC in the concentration scale strongly depends on alkyl radical length. All K(с) curves exhibit a third break, which corresponds to second critical micelle concentration CMC2, at which the properties of ionic-surfactant solutions may substantially change because of the appearance of supramicellar structures. The experimental data obtained have been used to evaluate the parameters of the model of electrical percolation for micellar solutions, i.e., effective conductivity \({\tilde K_m}\) and effective micelle radius r 0.

Electrical percolation in micellar sodium dodecyl sulfate solutions. a phenomenological consideration

Effects of electrical percolation accompanying variations in overall surfactant concentration с have been studied by the example of micellar sodium dodecyl sulfate solutions. It has been found that, in the studied concentration range of 0.001–1.2 M, dependences of electrical conductivity K on c may exhibit at least three break points, with the dK/dc derivatives changing in the vicinities of these points. At two of these points, which are reliably identified and correspond to critical micelle concentrations (CMC1 and CMC2), they decrease. At the third concentration, lying between CMC1 and CMC2, the dK/dc derivative increases. A substantiated assumption has been put forward that this break point, at which the dK/dc derivative increases, results from the clustering of micelles and the appearance of channels with a higher specific conductivity, which is provided by the contribution from the electrical conductivity of the diffuse and dense parts of micelle electrical double layers, upon the formation of clusters. The ionic surfactant concentration that corresponds to the break point at which the dK/dc value increases has been denoted as the critical percolation concentration.

Single-Walled Carbon Nanotubes in Highly Viscous Media: A Comparison between the Dispersive Agents [BMIM][BF4 ], L121, and Triton X-100.

Dispersions of single-walled carbon nanotubes (SWNTs) have been prepared by using the room-temperature ionic liquid [BMIM][BF4 ] (1-butyl-3-methylimidazolium tetrafluoroborate), the triblock copolymer Pluronic L121 [poly(ethylene oxide)5 -poly(propylene oxide)68 -poly(ethylene oxide)5 ] and the non-ionic surfactant Triton X-100 (TX100) in the pure state. The size of the SWNTs aggregates and the dispersion degree in the three viscous systems depend on the sonication time, as highlighted by UV/Vis/NIR spectroscopy and optical microscopy analysis. A nonlinear increase in conductivity can be observed as a function of the SWNTs loading, as suggested by electrochemical impedance spectroscopy. The generation of a three-dimensional network of SWNTs showing a viscoelastic gel-like behavior above a critical percolation concentration has been found at 25 °C in all the investigated systems by oscillatory rheology measurements.

Branched Wormlike Micelles Formed by Self-Assembled Comblike Amphiphilic Copolyelectrolytes

The structure of the self-assemblies formed by amphiphilic comblike copolyelectrolytes dispersed in water has been investigated by scattering techniques (light and neutron) and by transmission electronic microscopy. The comblike polymers consisted of a polystyrene backbone grafted with a fixed amount of pendant N,N-dimethyl quaternary ammonium alkyl groups of various lengths ranging from C12 up to C18. In aqueous solution, the polymers self-assembled into small spherical aggregates at low concentrations and into cylindrical aggregates above a critical concentration with a diameter that increased with the length of the alkyl side chains. The length of the cylindrical aggregates increased with increasing concentration, and branching occurred at higher concentration, which induced gelation above a critical percolation concentration. Growth and branching were favored by increasing the ionic strength of the solution. The dynamics slowed down with decreasing temperature and increasing alkyl length, and the asse...

The optical and electrical properties of silver nanowire mesh films

We present experimental results for the transmission T, haze H, sheet resistance Rs, and its spatial fluctuations ΔRs for silver nanowire films. Mie light scattering theory of nanowires is developed to predict both T and H as a function of diameter D of wires and the surface fraction ϕs covered by the wires. Percolation theory is used to derive an equation for Rs in terms of D, the aspect ratio of wires D/L and ϕs. The critical exponent t for percolation of Rs is found to be 1.23 in close agreement with theoretical results for 2D random resistive networks (t = 1.3). These equations show the importance of both the distributions of diameter ⟨D⟩ and aspect ratio of wires ⟨D⟩⟨L⟩/⟨L2⟩ to predict the optical and electrical properties. Spatial fluctuations ΔRs/Rs can also be significant in these films and be greater than 10% as ϕs approaches the critical percolation concentration ϕc. We show that the calculated T versus Rs and H versus Rs curves are in good agreement with the experimental data. We propose figures of merit for percolating nanowire films in terms of high T, low H, and low Rs to order the quality of films for touch screen applications. The results show that D < 50 nm and L > 5 μm are needed to achieve low haze H < 1%, high transmission T > 90%, together with low Rs ∼ 100 Ω/sq for touch screen applications. Finally, we present experimental and theoretical results of the real and imaginary refractive indices of AgNW/polymer nanocomposites, and find that the Van De Hulst model is more accurate than the Maxwell Garnett models.

Slow dynamics in transient polyelectrolyte hydrogels formed by self-assembly of block copolymers

Transient polyelectrolyte hydrogels were formed by self-assembly of triblock copolyelectrolytes with a central hydrophilic block, poly(acrylic acid) (PAA), and two hydrophobic end blocks, poly(n-butyl acrylate(50%)-stat-acrylic acid(50%)) [P(nBA(50%)-stat-AA(50%))]. The relaxation of the concentration fluctuations was investigated by dynamic light scattering as a function of the concentration, the pH, the temperature, and the ionic strength. A relatively fast mode was observed at all polymer concentrations caused by cooperative diffusion of the polymers. Above the critical percolation concentration a second slow relaxation mode was observed caused by a linear displacement of small heterogeneities in the network with constant velocity. The relative amplitude of the slow mode was determined by the strength of the electrostatic repulsion. The velocity of the displacement in the transient network is shown to be directly correlated to the terminal relaxation time of the shear modulus and has the same Arrhenius temperature dependence. Both the velocity of the displacement and the mechanical relaxation strongly slow down with decreasing degree of ionization below 0.7 and increasing ionic strength above 0.5 M. A ballistic relaxation process has been reported earlier for colloidal gels, and the present study shows that it can also occur in polymeric networks.

Berezinskii–Kosterlitz–Thouless transition close to the percolation threshold

Abstract We use Monte Carlo to investigate the Berezinskii–Kosterlitz–Thouless transition close to the site percolation threshold in a square lattice. Several thermodynamic quantities are calculated for lattice sizes L × L , from 16 L 640 . Our results are consistent with an infinite order transition for any value of the concentration of magnetic sites. We found that close to the critical percolation concentration, p c (0.592746), the Berezinskii–Kosterlitz–Thouless transition temperature goes to zero as T BKT ∝ ( p − p c ) 0.908 and the specific heat behaves as T s h ∝ p 1.133 .

Morphology, microstructure, and electrical properties of poly(D,L‐lactic acid)/carbon nanocapsule composite nanofibers

AbstractThe morphologies and properties of poly(D,L‐lactic acid)/carbon nanocapsule (CNCs) electrospun fiber mats were studied. To determine the effect of filler, a solution without sufficient entanglement density was used. The rheological results showed that the solution entanglement density was enhanced by the addition of filler, which led to the formation of network between the polymer chains and filler particles. The findings can be used for further exploration of electrospinning in semidilute solutions. The electrical conductivities of electrospun fiber mats and cast films were determined and fitted using the percolation theory. When the CNC content was above the critical percolation concentration, a master curve was obtained provided that the conductivity curves were normalized with the level of percolation network. The porosity effect was investigated. In addition, the microstructure of as‐spun composite fiber was examined in detail. © 2012 Wiley Periodicals, Inc. J. Appl. Polym. Sci., 2013

Structure and Rheology of Self-Assembled Telechelic Associative Polymers in Aqueous Solution before and after Photo-Cross-Linking

Poly(ethylene oxide) (PEO) was end-capped with short poly(methacryloyloxyethyl acrylate) (PMEA) chains. In aqueous solutions the end-blocks self-assembled, leading to flower-like polymeric micelles at low concentrations that associated through bridging at higher concentrations. Above a critical percolation concentration (Cp) a system-spanning transient network was formed. The transient self-assemblies were locked-in in situ by UV irradiation which caused photo-cross-linking of the PMEA blocks. The structure and composition of the clusters formed at C < Cp was studied by light scattering and chromatography after cross-linking and dilution. The results were compared with mean-field theory and numerical simulations. The dynamic mechanical properties of the networks formed for C > Cp were studied by oscillatory shear before, during, and after irradiation. UV radiation transformed the transient networks into permanent gels within a few seconds.

Conformational behaviors of syndiotactic polystyrene in chloroform revealed by intrinsic fluorescence technique

Conformational behaviors of syndiotactic polystyrene (sPS) in chloroform were investigated by detecting its concentration dependence of intrinsic fluorescence in the range from 25 to −62 °C. For the dimer emission, the coexistence of T-shaped configuration with sandwich configuration was observed and the contribution of sandwich configuration increases with increasing sPS concentration in the system. In addition, three conformations are observed in sPS/CHCl 3 systems: (I) coil phase; (II) coil-to-helix transformation; (III) helix-to-network transformation. The conformational ordering of polymer chain in super dilute sPS/CHCl 3 systems during isothermal process indicated that micro-gel formed when the concentration was higher than 0.2 g L −1, and single helix formed at 0.09 ∼ 0.2 g L −1 proceeds through a unimolecular or self-organization mechanism and no helical aggregates were found. The helix was formed by clustering of several chain segments when the polymer concentration was below 0.09 g L −1. Moreover, a model was proposed to describe the elasticity of sPS/CHCl 3 systems near and beyond the critical percolation concentration.

DC electrical conductivity of carbon black polymer composites at low temperatures

The understanding of the charge transport mechanism of the conducting polymers and their composites is important for the successful fabrication of semiconductor devices. In this work, the DC electrical conductivity of carbon black particles in a polystyrene matrix, in the temperature range of 80 to 250 K is examined, for concentrations below the critical percolation concentration. We show that three-dimensional variable range hopping is observed, that is, the temperature behavior of the DC conductivity can be expressed by σ = σ 0 exp − T 0 T 1 4 . From the hopping parameter we obtained the mean hopping distance, the density of active centers and the energy associated with it.

Quasi-Two-Dimensional Network Structures Formed by Tin Oxide Nanoparticles at Air/Solid interfaces

We study the microstructure of a recently developed composite film [A. Wakabayashi et al.: Langmuir 22 (2006) 9260] by field emission scanning electron microscopy and by applying the box-counting method to the obtained microscope images. The fractal analysis of the images revealed that a fractal network structure consisting of antimony-doped tin oxide (ATO) nanoparticles was formed on the film surface. The identified microstructure is consistent with the resistivity behavior of a film with a low critical concentration for percolation.

Dielectric properties of the ethylene butylacrylate/carbon black nanocomposites

In this paper we report the d.c. electrical conductivity measurements between 80 and 350 K, and the dielectric properties at low frequencies (10 Hz < f < 100 kHz) at constant temperature T = 300 K, of the ethylene butylacrylate/carbon black nanocomposites, with different concentrations of the dispersed conducting particles. The concentration of the conductive inclusions has been proved to be a crucial parameter, governing the electrical behavior of these nanocomposites. Above the critical percolation concentration, that is 12% of conducting particles, the Cole–Cole model of dielectric relaxation can be used to interpret the obtained data. The calculated relaxation parameters of the model show that, with the increasing concentration of carbon black nanoparticles, the system becomes more heterogeneous, which is confirmed by the broader distribution of the relaxation times.