Computer simulation of fractal dimensions of fat crystal networks

The rheological properties of fat-structured products are determined by the microstructure of their fat crystal networks, which can be quantified by using microscopical and rheological techniques. Of particular interest to this study is the quantification of the fractal dimension of the network using these two techniques. Fractal dimensions determined by polarized light microscopy include box-counting, particle-counting, and Fouriertransform fractal dimensions, whereas the fractal dimensions determined by small deformation dynamic rheology exploit the dependence of the storage modulus on the solids' volume fraction. This work reveals that different microscopy fractal dimensions are sensitive to different microstructural factors within the fat crystal network, and thus have different physical meanings. The boxcounting fractal dimension, D _b , increases with increases in crystal size and area fraction of the fat crystals, whereas the particlecounting fractal dimension, D _f , is sensitive to the radial distribution pattern of fat crystals; and the Fourier-transform fractal dimension, D _FT, decreases with increasing crystal size. In the studies on the macroscopic physical properties of fat crystal networks, it is necessary to find the determining structural characteristics and then use the fractal dimensions that are most closely related.     

Effect of microgel structure on epoxy polymer curing in the presence of carbon nanotubes

The catalytic effect of carbon nanotubes on epoxy polymer curing has been studied using the fractal method. The fractal dimension of epoxy microgels has been shown to be reduced by the interaction with the surface of carbon nanotubes. This effect is the only factor that contributes to the acceleration of epoxy crosslinking.     

Morphological and fractal studies of polypropylene/poly(ethene‐1‐octene) blends during melt mixing using scanning electron microscopy

BACKGROUND: Polymer blending creates new materials with enhanced mechanical, chemical or optical properties, with the exact properties being determined by the type of morphology and the phase dimension of the blend. In order to control blend properties, morphology development during processing needs to be understood. The formation and evolution of polypropylene/poly(ethylene‐1‐octene) (PP/POE) blend morphology during blending are qualitatively represented by a series of time‐dependent scanning electron microscopy (SEM) patterns. The area diameter and its distribution of dispersed phase domains are discussed in detail. In order to characterize the formation and evolution of phase morphology quantitatively, two fractal dimensions, D_s and D_d, and their corresponding scaling functions are introduced to analyze the SEM patterns. RESULTS: The evolution of the area diameter indicates that the major reduction in phase domain size occurs during the initial stage of melt mixing, and the domain sizes show an increasing trend due to coalescence with increasing mixing times. The distribution in dispersed phase dimension obeys a log‐normal distribution, and the two fractal dimensions are effective to describe the phase morphology: D_s for dispersed phase dimension and D_d for the distribution in it. CONCLUSIONS: The fractal dimensions D_s and D_d can be used quantitatively to characterize the evolutional self‐similarity of phase morphology and the competition of breakup and coalescence of dispersed phase domains. It is shown that the fractal dimensions and scaling laws are useful to describe the phase morphology development at various mixing times to a certain extent. Copyright © 2007 Society of Chemical Industry     

Fractal Dimensions of Simulated and Real Fat Crystal Networks in 3D Space

The microstructure of fat crystal networks is closely related to rheological properties of fat products, and thus is of particular interest to food scientists. The fractal dimensions of fat crystal networks calculated by microscopy methods such as box-counting, D _b, particle-counting, D _f, and mass fractal dimension, D _m, have been extensively employed to quantify the microstructure of fats. This work revealed the microstructural basis of D _b, D _f, and D _m in 3D space through both computer simulation and experiments on the high melting fraction of milk fat crystal networks. Similar to our previous simulation study on the fractal dimensions of fat crystal networks in 2D space, D _b is sensitive to crystal size, area fraction, and not sensitive to distribution orderliness of crystals, which is the percentage of evenly distributed fat crystals within the simulated fat crystal networks. D _f and D _m were not affected by any of the microstructural factors studied in the simulation.     

Determination of the fractal dimension of alumina using adsorption: Challenges of choosing appropriate adsorbates and analysis methods

The classical method for surface characterisation of the porous structure of the catalysts is nitrogen (N₂) adsorption at ?196°C, which provides a catalyst surface area value valid for molecules similar in size to N₂ (0.162 nm²/molecule). To complement and obtain more information about the materials, catalyst surfaces can be characterised using fractal geometry. The fractal dimension of a sample can be determined by the adsorption uptake of molecules of different sizes on the surface of interest in order to obtain a characteristic parameter of the surface geometry known as the fractal dimension, D. In this work, the value of D for a γ‐alumina catalyst support has been determined (D = 2.71 ± 0.15) using different adsorbates (argon, nitrogen, isopropanol, pyridine, and n‐butane). The decision process for choosing these adsorbates and the challenges of this type of characterisation method are discussed in this article. © 2011 Canadian Society for Chemical Engineering     

Dependence of the Mechanical Sensitivity on the Fractal Characteristics of Octahydro‐1,3,5,7‐tetranitro‐1,3,5,7‐tetrazocine Particles

Three fabrication methods were used to synthesize HMX powders with different particle sizes and microscopic morphologies. All as‐prepared samples were characterized by laser granularity measurements and scanning electron microscopy (SEM). The mechanical sensitivity and thermal stability of the different HMX powders were characterized using mechanical sensitivity tests and differential scanning calorimetry (DSC). Size distribution data and SEM images were used to find the size fractal dimension (D) and surface fractal dimension (D_s) of HMX samples, which were calculated by the least‐squares method and fractal image processing software (FIPS), respectively. The parameters D and D_s quantize two important properties of HMX particles, namely the complexity of the particle size distribution and the irregularity of the particle surface, which affect the thermal conductivity of the particle group if it is exposed to stimuli such as impact, friction or heating. The fractal dimensions reveal the dependence of the mechanical sensitivity of HMX on the powder size, size distribution and microscopic morphology. The results indicate that the proportion of fine particles in HMX powder increases as the D value increases, which causes decreased impact sensitivity. This occurs because hot spot formation leading to an explosion is more difficult because of the improved thermal conductivity of the particle group. Similarly, the surface roughness of HMX particles increases with an increase in D_s, causing an increase in friction sensitivity because of the excessive accumulation of frictional heat. In addition, thermal analysis results indicate that the maximum thermal decomposition rate of HMX decreases with increasing D and D_s.     

CO2载体CaO循环煅烧/碳酸化反应的分形特征

捕捉煤燃烧释放出的CO₂时,作为CO₂载体的CaO微观结构特性对其循环碳酸化性能具有显著影响。采用分形维数作为表征CaO微观结构的特征参数,研究在循环煅烧/碳酸化反应过程中CaO的分形特征及其对CO₂捕捉性能的影响规律。结果表明,随着循环次数的增加CaO分形维数逐渐下降,CaO孔道也由粗糙和不规则变得越来越平滑和有规则性。煅烧温度升高则CaO分形维数下降。分形维数较大的CaO具有较高的碳酸化速率。在碳酸化过程的前10 min内CaO的分形维数迅速减小,此后随时间变化缓慢。在分形维数D≤2.61的实验范围内,CaO分形维数与其循环碳酸化转化率呈线性正相关;当D＞2.61时,可能存在临界分形维数D_cr,当D＞D_cr时随着分形维数的进一步增大CaO转化率反而减小。     

Fractal analysis of polypropylene composite filled with nano‐calcium carbonate

Polypropylene (PP) and nano‐calcium carbonate (CaCO₃) composites were prepared by melt mixing in a corotating twin‐screw extruder. Transmission electron microscopy study and particle size analysis revealed the dispersion and the size distribution of CaCO₃ in PP. With the increase of loading of filler, CaCO₃ nanoparticles densely aggregated together and the dispersion of filler became bad. The fractal dimensions of the composites were determined using fractal concept. The fractal dimensions of D and D_k described the irregularities of the shape of an object and the distributions of particle populations, respectively. The D and D_k values were influenced by the content of filler, i.e., the D values increased, and the D_k values decreased with the increase of loading of filler. When the loading of filler was low, the values of D and D_k of PP composites differ slightly than the counterparts of PP/PP‐g‐MA (50 wt %) blend. For 20 wt %, they were almost identical. This fact showed that the fractal dimension was correlated with the dispersion. © 2008 Wiley Periodicals, Inc. J Appl Polym Sci, 2008     

Reactive blends of epoxy resin (DGEBA) crosslinked by anionically polymerized polycaprolactam: Process of epoxy cure and kinetics of decomposition

The curing reactions, kinetics, morphology, and thermal stability of the reactive blends of diglycidyl ether of bisphenol‐A (DGEBA) and polycaprolactam were studied by differential scanning calorimetry (DSC), Fourier transform infrared spectroscopy (FTIR), scanning electron microscopy (SEM), and thermogravimetric analysis. DSC studies showed that the heat of reaction (ΔH) increased when the DGEBA content was increased from 50 to 80 wt % and increased drastically above 70 wt % DGEBA content because of an increase in the extent of crosslinking. The activation energy and pre‐exponential factor of cure reactions increased drastically with an increase in the DGEBA content above 70 wt % because of a drastic increase in crosslink density. The extent of curing reaction of polycaprolactam with DGEBA is dependent on the blend composition. The nucleophilic attack on oxirane ring by amide nitrogen of polycaprolactam is a dominant curing reaction in low DGEBA compositions, and another type of curing reaction with relatively large activation energy and pre‐exponential factor also occurred, which becomes dominant when the DGEBA content reaches above 70 wt %. FTIR studies also revealed that two types of reactions do exist during the curing of polycaprolactam with DGEBA. It was observed during SEM studies that the reactive blends show multiphase system and on increasing the DGEBA content from 50 to 80 wt %, the mixing of the two phases increased. The reactive blend Ep₈₀Ca₂₀ with 80 wt % DGEBA content exhibits a single‐phase system because of better mixing of the two phases. The results of thermogravimetric analysis also indicate that the initial degradation temperature (T_i), activation energy (E), and pre‐exponential factor (Z) increased with increasing DGEBA content from 50 to 80 wt % in the reactive blends and increased drastically above 70 wt % DGEBA content due to the higher crosslink density. © 2004 Wiley Periodicals, Inc. J Appl Polym Sci 92: 687–697, 2004     

Photon transmission technique for monitoring formation and swelling of polyacrylamide gels

The in situ, real-time photon transmission technique was used to monitor the free radical crosslinking copolymerization of acrylamide and N,N′-methylenebisacrylamide (Bis). Gelation experiments were performed with various Bis contents at various wavelengths. It was observed that the transmitted photon intensity, I _tr, decreased dramatically at a certain reaction time, which is attributed to the increase in scattered light intensity, I _sc, during the formation of microgels in the system. The increase in I _scwas modeled using Rayleigh's equation where the reaction time was found to be proportional to the volume of the microgels. The disc-shaped polyacrylamide (PAAm) gels were dried before use during swelling experiments. Transmitted light intensity, I _tr, from the gel increased at initial stages when PAAm gels were immersed in water and then decreased exponentially as the swelling time increased. Decrease in I _trwas attributed to the lattice heterogeneities, which might have originated between microgels and holes in the swelling gel. Decrease in I _trwas modeled using the Li–Tanaka equation from which cooperative diffusion coefficients, D _c, were determined for gels of various Bis contents. It is observed that the D _cvalues increased with the Bis content.     

Polymerization in a magnetic field. 14. Possibilities to improve field effect during methyl acrylate polymerization

Magnetokinetic changes intervened in the radical polymerization of methyl acrylate in a continuous external magnetic field of 0.22 T was evidenced. The use of a matrix of polymerization, respectively, poly(ethylene glycol), was correlated with the obtained kinetic data. The occurring‐magnetokinetic changes were justified by the radical pair mechanism, Δg, which corresponds to the benzoyl peroxide decomposition into a magnetic field. The intervened magnetokinetic modifications during the reactions are correlated with the reaction conditions. The influence of the magnetic field is confirmed by the molecular weights of the obtained polymers, values interpreted from the viewpoint of the reaction conditions. © 2004 Wiley Periodicals, Inc. J Appl Polym Sci 92: 1031–1036, 2004     

Influence of ultrasonic treatment in PP/CNT composites using masterbatch dilution method

Composites of polypropylene (PP) filled with multiwalled carbon nanotubes (CNTs) of various concentrations were prepared by a twin screw extruder using direct compounding (DC) method without and with ultrasonic treatment. In addition, a masterbatch of 20 wt% PP/CNT composites were prepared without and with ultrasonic treatment and diluted to the same concentrations as in the DC method without ultrasonic treatment. This is called the masterbatch dilution (MD) method. The rheological, electrical and mechanical properties were investigated. The microdispersion was determined using optical microscopy to correlate the processing, properties and structure. It was shown that the MD method provided better dispersion of CNT in PP matrix than the DC method. This was indicated by an increase of the storage modulus, viscosity, electrical and mechanical properties. The fractal dimension of CNTs, D, and the backbone fractal dimension, x, of the CNT network were determined by fitting the rheological data to the scaling model. The lower fractal dimension of CNT and higher backbone fractal dimension of CNT network in composites prepared by the MD method compared with composites obtained by the DC method indicated a better dispersion. Additionally, a lower D and a higher x values as well as the favorable effect on the morphology and mechanical properties were achieved when the ultrasonic treatment at an amplitude of 13 μm was applied in the MD method, indicating an advantage in use of the ultrasonic treatment in preparing the PP/CNT masterbatch.     

Fractal characteristic of three Chinese coals

Experimental and theoretical investigation about coal/char structure is presented. Surface structures of parent coal and char with different burn-off ratios were analyzed. We introduced the fractal theory into Scanning Electron Microscopy image analysis and utilize the particle surface fractal dimension (D_ps) to quantitatively describe the surface character of coal/char particles. D_ps of three Chinese coals reach their maximum in the 35-45 wt% char burn-off interval and then decrease with increasing carbon burn-off ratio. The inner-pore information of coal/char particles was determined by N₂ isotherm adsorption/desorption. Using fractal BET model, internal surface fractal dimension (D_s) of coal/char particles was calculated. The D_s change trend of three Chinese coals is similar to their S_BET development. It means the D_s can quantitatively describe the inner pore structure character of coal/char particles.     

Dynamic modeling of curing process of epoxy prepreg

The dynamic curing process was studied by using differential scanning calorimetry (DSC) and modeled by two methods. One was based on the Kissinger and Ozawa approach, in which the activation energy was taken as a constant for all the heating rates. The whole curing process was modeled with two cure reactions. Reaction 1 exhibited the behavior of the autocatalytic reaction, whereas Reaction 2 was the nth order reaction. The effect of heating rate on the preexponential factor A₁ of Reaction 1 was apparent. As the heating rate increased, the A₁ decreased. There was no significant effect of heating rate on the preexponential factor A₂ of Reaction 2 and the reaction orders for both reactions. The calculated results showed that the contributions of these two reactions to the total curing process were very different and changed with the heating rate. Except in the early cure stage, the calculated total degree of cure agreed well with the experimental data. Another method was based on the Borchardt and Daniels kinetic approach, where the activation energy of the cure reaction at each heating rate was determined separately. The whole curing process was modeled with one autocatalytic reaction. The fitting results showed that both preexponential factor and activation energy increased with the increment of the heating rate. As in the first method, the effect of heating rate on the orders of reaction was very small. The calculated results agreed well with experimental values in the early cure stage. © 2002 Wiley Periodicals, Inc. J Appl Polym Sci 86: 1911–1923, 2002     

The fractal and scaling analysis of chemical reactions

It is shown that the geometrical factor controlling a thermooxidative degradation course is effective spectral dimension d or factor β. In turn, parameters d and β are a function of a polymer melt structure characterized by its fractal dimension Δ_f. The applicability of this or that scaling relationship allows the evaluation of general features of chemical reactions in the thermooxidative degradation process. © 2004 Wiley Periodicals, Inc. J Appl Polym Sci 93: 2343–2347, 2004     

Phenolic foams from wood tar resols

This article investigates resole‐type phenolic foams, in which 40% phenol was substituted with distilled oily fractions from wood tar. Different proportions of blowing agent, hexamethylenetetramine, and stabilizer, a sodium alkyl sulfate surface active agent, were used to produce a series of rigid foams, which were characterized by curing index, density and hardness. The average curing index was about 90% and densities ranged from 480 to 960 kg.m^?3. A fractal dimension (D_f) approach was used to assess the homogeneity of pores within the foams and an average (dimensionless) value of 2.6 was found, indicating a non uniform structure. Compressive strength tests of the foams gave values above 1.0 MPa, which is higher than those for typical phenolic foams. © 2009 Wiley Periodicals, Inc. J Appl Polym Sci, 2010     

Novel Cross-Linking Mechanism with Diol Type Cross-Linkers,to Prepare PAA Microgels via Precipitation Polymerization Method

This article describes a new cross-linking mechanism for synthesizing PAA microgels by precipitation polymerization method. The post curing stage was defined, for the first time, in this study. Effect of cross-linker concentration, and the time and temperature of post curing stage on the samples properties was investigated. A new relationship between gelation rate in the post curing stage and hydroxyl content was defined. Also a new relationship between M_c, hydroxyl concentration and temperature of post curing stage was suggested too. According to rheological characterization, microgels synthesized by novel mechanism have better rheological properties than microgels synthesized by conventional mechanism.     

The fractal analysis of copolymerization processes

The results obtained in the present article show that the fractal analysis and irreversible aggregation models application allow to obtain a clear physical picture of copolycondensation process and estimate its quantitative characteristics. The basic characteristic, controlling this process, is fractal dimension D_f of macromolecular coil in solution. © 2008 Wiley Periodicals, Inc. J Appl Polym Sci, 2009     

Curing kinetics and thermal property characterization of an o‐cresol formaldehyde epoxy resin and 4,4′‐diaminodiphenyl ether system

The kinetics of the curing reaction for a system of o‐cresol formaldehyde epoxy resin (o‐CFER) with 4,4′‐diaminodiphenyl ether (DDE) as a curing agent were investigated with differential scanning calorimetry (DSC). An analysis of the DSC data indicated that an autocatalytic behavior appeared in the first stages of the cure for the system, and this could be well described by the model proposed by Kamal, which includes two rate constants and two reaction orders (m and n). The overall reaction order (m + n) was 2.7–3.1, and the activation energies were 66.79 and 49.29 kJ mol^?1, respectively. In the later stages, a crosslinked network was formed, and the reaction was mainly controlled by diffusion. For a more precise consideration of the diffusion effect, a diffusion factor was added to Kamal's equation. In this way, the curing kinetics were predicted well over the entire range of conversions, covering both the previtrification and postvitrification stages. The glass‐transition temperatures of the o‐CFER/DDE samples were determined via torsional braid analysis. The results showed that the glass‐transition temperatures increased with the curing temperature and conversion up to a constant value of approximately 370 K. The thermal degradation kinetics of the system were investigated with thermogravimetric analysis, which revealed two decomposition steps. © 2004 Wiley Periodicals, Inc. J Appl Polym Sci 94: 182–188, 2004