Studies of physico-chemical properties and fractal dimensions of selected high-temperature superconductor surfaces

Properties relating to porosity of solids (fractal dimensions, surface roughness parameters) were evaluated from atomic force microscopy (AFM) and nitrogen adsorption-desorption isotherms measured at 77 K for selected high-temperature [(RE) Ba₂Cu₃O_7−x, RE=Y, Sm] superconductors. Adsorption capacity, specific surface area, fractal dimensions were determined from adsorption-desorption isotherms. The adsorption isotherms of all samples were S-shaped and belong to type II according to the IUPAC classification. A linear relationship was demonstrated between the fractal coefficients calculated by using the two methods and values of adsorption capacity of monolayer.     

Electrochemical technique for the determination of fractal dimension of dental surfaces

Fractal dimension of a carious tooth surface was determined using an electrochemical method. The method was based on time-dependent diffusion towards electrode surfaces, which is one of the most useful and reliable methods for the determination of fractal dimension of electrode surfaces. For this purpose, the tooth was covered with a gold layer, which acted as an electrode in electrochemical experiments. It is suggested that the fractal dimension can be used as a quantitative measure of the state of dental surfaces. The method presented demonstrates the power of electrochemical techniques for the determination of fractal dimension of surface of non-conducting objects.     

Morphology of dry lignins and size and shape of dissolved kraft lignin particles by X-ray scattering

Lignin is a highly branched polymer consisting of phenylpropane units, and it is one of the ingredients of the supporting matrix in plant cell walls. The morphology of several lignins extracted from plant cell walls using different methods was studied by small-angle and ultra-small-angle X-ray scattering. A power-law type intensity was observed for the dry lignins, but on the basis of the power-law exponent the fractal approach often applied to lignins is not fully justified. However, the intensity of kraft lignin did show a power law with surface fractal dimension D(s) = 2.7 +/- 0.1. The specific surface area of the lignins ranged from about 0.5 to 60 m(2)/g with 20% relative accuracy. The radius of gyration was determined from small-angle X-ray scattering data for aqueous solutions of kraft lignin. The shape of the particles in NaCl and NaOH solutions was found to be elongated. The particles were about 1-3 nm thick, while the length (5-9 nm) depended on the solvent and on the lignin concentration. The size of these primary particles was approximately the same as the size of the pores in the fractal aggregates of the dry kraft lignin. Their size was determined to be about 3.5 nm.     

A fractal analysis of analyte-estrogen receptor binding and dissociation kinetics using biosensors: environmental effects

A fractal analysis is used to model the binding and dissociation kinetics between analytes in solution and estrogen receptors (ER) immobilized on a sensor chip of a surface plasmon resonance (SPR) biosensor. Both cases are analyzed: unliganded as well as liganded. The influence of different ligands is also analyzed. A better understanding of the kinetics provides physical insights into the interactions and suggests means by which appropriate interactions (to promote correct signaling) and inappropriate interactions such as with xenoestrogens (to minimize inappropriate signaling and signaling deleterious to health) may be better controlled. The fractal approach is applied to analyte-ER interaction data available in the literature. Numerical values obtained for the binding and the dissociation rate coefficients are linked to the degree of roughness or heterogeneity (fractal dimension, D(f)) present on the biosensor chip surface. In general, the binding and the dissociation rate coefficients are very sensitive to the degree of heterogeneity on the surface. For example, the binding rate coefficient, k, exhibits a 4.60 order of dependence on the fractal dimension, D(f), for the binding of unliganded and liganded VDR mixed with GST-RXR in solution to Spp-1 VDRE (1,25-dihydroxyvitamin D(3) receptor element) DNA immobilized on a sensor chip surface (Cheskis and Freedman, Biochemistry 35 (1996) 3300-3318). A single-fractal analysis is adequate in some cases. In others (that exhibit complexities in the binding or the dissociation curves) a dual-fractal analysis is required to obtain a better fit. A predictive relationship is also presented for the ratio K(A)(=k/k(d)) as a function of the ratio of the fractal dimensions (D(f)/D(fd)). This has biomedical and environmental implications in that the dissociation and binding rate coefficients may be used to alleviate deleterious effects or enhance beneficial effects by selective modulation of the surface. The K(A) exhibits a 112-order dependence on the ratio of the fractal dimensions for the ligand effects on VDR-RXR interaction with specific DNA.     

World of nanostructures - nanotechnology,surface properties of chosen nanomaterials

Summary The paper presents the basic information on nanotechnology and the recent results of studies of physicochemical properties of chosen nanomaterial surfaces (montmorillonites, carbon nanotubes, smart surfaces) by means of complex measuring methods. Physicochemical properties of nanomaterial surfaces by means of the special thermogravimetry Q-TG, sorptometry, porosimetry, atomic force microscopy (AFM) and scanning electron micrograph (SEM) methods were investigated. A numerical and analytical procedure for the evaluation of total heterogeneous properties (desorption energy distribution and pore-size distribution functions) on the basis of liquid thermodesorption from the sample surfaces under the quasi-equilibrium conditions and sorptometry techniques are presented. The evaluation of the fractal dimensions of nanotubes using the sorptometry, porosimetry, thermogravimetry Q-TG and AFM data are presented. The comparison of fractal coefficients calculated based on them with the results from Q-TG, sorptometry, porosimetry and AFM gave good agreement.     

A beta-cyclodextrin based fiber-optic chemical sensor: a fractal analysis

A fractal analysis is presented for the binding of pyrene in solution to beta-cyclodextrin attached to a fiber-optic chemical sensor. The specific (k(l)) and non-specific binding rate coefficients and the fractal dimension (D(f)) (specific binding case only) both tend to increase as the pyrene concentration in solution increases from 12.4 to 124 ng ml(-1). Predictive relations for the binding rate coefficient (specific as well as non-specific binding) and for D(f) (specific binding case only) as a function of pyrene concentration are provided. These relations fit the calculated k(l) and D(f) values in the pyrene concentration range reasonably well. Fractal analysis data seem to indicate that an increase in the pyrene concentration in solution increases the "ruggedness" or inhomogeneity on the fiber-optic biosensor surface. The fractal analysis provides novel physical insights into the reactions occuring on the fiber-optic chemical surface and should assist in the design of fiber-optic chemical sensors.     

Determination of the textural characteristics of carbon samples using scanning electronic microscopy images: comparison with mercury porosimetry data

In this paper, scanning electron microscopy (SEM) images of carbon samples were analysed, and some textural characteristics were obtained and compared with those determined using mercury porosimetry data. Fractal dimensions were calculated from both mercury porosimetry and SEM images as methods for characterising the porous distribution (heterogeneity) of the samples. Lacunarity is easily determined from SEM images as a measure of the degree of heterogeneity of a porous surface. A relationship between the lacunarity and the fractal dimensions calculated using both methods is shown. Pore-size distributions were also determined from the analysis of SEM images. We show that the analysis of SEM images is a valuable complement to mercury porosimetry measurements and a useful tool for the characterisation of porous surfaces. This method offers the possibility of evaluating the features of porous materials and comparing the results to those obtained using mercury intrusion analysis.     

Studies of physico-chemical properties and fractal dimensions of MgB2 superconductor surface

The porous structure of MgB₂ has been investigated using atomic force microscopy (AFM) and sorption techniques. The fractal dimension and surface roughness parameters were evaluated from (AFM) and nitrogen adsorption?Cdesorption isotherms measured at ?196?°C for MgB₂ sample. Adsorption capacity, specific surface area, and fractal dimensions were determined from adsorption?Cdesorption isotherms. The sorption isotherms of MgB₂ samples were S-shaped and belong to type II according to the IUPAC classification. The results of fractal dimensions of MgB₂ surface determined on the basis sorptometry and AFM data are compared.     

Surface analysis of cryofixation-vacuum-freeze-dried polyaluminum chloride-humic acid (PACl-HA) flocs

The powder of polyaluminum chloride-humic acid (PACl-HA) flocs was prepared by cryofixation-vacuum-freeze-drying method. The FTIR spectra show that some characteristic functional groups in polyaluminum chloride (PACl), humic acid (HA), and kaolin still existed in the dried flocs. X-ray diffractometry (XRD) patterns indicate that these flocs are amorphous. Nitrogen adsorption-desorption isotherms were obtained for different samples of the dried PACl-HA flocs. The BET specific surface area, BJH cumulative absorbed volume and BJH desorption average pore diameter of them were determined. The peak values of 8.4-11.2 nm (pore diameter) for pore size distribution (PSD) curves indicate that the pores of the dried flocs are mostly mesopores. The surface fractal dimensions D(s) and the corresponding fractal scales determined from both SEM images and nitrogen adsorption-desorption data sets reveal the multi-scale surface fractal properties of the dried PACl-HA flocs, which exhibited two distinct fractal regimes: a regime of low fractal dimensions (2.07-2.26) at higher scales (23-387 nm), mainly belonging to exterior surface scales, and a higher fractal dimensions (2.24-2.37) at lower scales (0.80-7.81 nm), falling in pore surface scales. Both HA addition and kaolin reduction in dried floc can decrease the irregularity and roughness of external surface. However, for the irregularity and roughness of pore surface, the addition of HA or kaolin in dried floc can increase them. Furthermore, some difference was found between the pore surface fractal dimensions D(s) calculated from nitrogen adsorption and desorption data. The pore surface D(s) values calculated through thermodynamic model were much greater than three.     

Measurement of Fractal Aggregates of Polydisperse Particles Using Small-Angle Light Scattering

The effect of primary particle polydispersity on the structure of fractal aggregates has been investigated through the salt-induced, diffusion-limited aggregation of mixtures of hematite. The fractal dimension was determined experimentally using three independent methods: q dependence of static light scattering, kinetic scaling, and correlation of aggregate mass and linear size both determined from Guinier scattering. The fractal dimensions D(f) obtained were 1.75+/-0.03, 1.76+/-0.03, and 1.70+/-0.05, respectively. The use of a previously derived fractal mean particle size was validated in allowing data collapse to master curves for the aggregation kinetics data. The fractal mean particle size is shown to have general utility by taking a number weighting to describe polydisperse aggregation kinetics and a mass weighting to describe small q scattering behavior. Copyright 2000 Academic Press.     

用吸附数据对硅胶表面的分形分析

根据一种中孔硅胶对甲酸、乙酸、丙酸和四氯化碳蒸气吸附等温线的单层区域和毛细凝结区域的数据以及自四氯化碳稀溶液中吸附系列脂肪醇的结果计算了该硅胶表面分形维数D.除了由吸附四氯化碳等温线毛细凝结区域所得D值低于2外,由其它吸附数据得出的该硅胶的分维D近似相等(D=2.06±0.05).中孔硅胶低D值可用其孔结构特性解释.     

Influence of drying procedure on the mesoporosity and surface fractal dimensions of silica xerogels prepared with different agitation methods

Silica xerogels were prepared by thermal drying wet gels in an electric oven (70 degrees C) after certain duration of ambient drying, and the relevant effect is investigated on the mesopore structures and surface fractal dimensions of the resultant xerogels. The silica gels were derived from a hydrochloric acid-catalyzed TEOS (tetraethylorthaosilicate) system, and both magnetic stirring and ultrasonic vibration were adopted during sol preparation. The percentage mesoporosity and surface fractal dimensions are evaluated using image analysis methods, based on FE-SEM (field emission gun-scanning electron microscopy) images. The results show that the mesoporosity of the resultant xerogels decreases with the duration of ambient drying for samples prepared using magnetic stirring and low-intensity ultrasonic vibration, while samples subjected to high-intensity ultrasound show a somewhat reverse trend. Samples prepared with magnetic stirring have almost constant surface fractal dimensions (nearly 3), irrespective of the ambient drying before thermal drying. The surface fractal dimensions of samples prepared using ultrasound increase with the duration of ambient drying.     

Topological derivation of shape exponents for stretched exponential relaxation

In homogeneous (ideal) glasses, the important dimensionless stretched-exponential shape parameter beta is described by magic (not adjusted) simple fractions derived from fractal configuration spaces of effective dimension d* determined by different topological axioms (rules) in the presence (absence) of a forcing electric field. The rules are based on a new central principle for defining glassy states: equal a priori distributions of fractal residual configurational entropy. Our approach and its beta estimates are fully supported by the results of relaxation measurements involving many different glassy materials and probe methods. The present unique topological predictions for beta typically agree with observed values to approximately 1% and indicate that for field-forced conditions beta should be constant for appreciable ranges of such exogenous variables as temperature and ionic concentration, as indeed observed using appropriate frequency-domain data analysis. The present approach can also be inverted and used to test sample homogeneity and quality.     

Surface heterogeneity and surface area from linear inverse gas chromatography. Application to glass fibers

A new method of measuring surface heterogeneity of non-porous adsorbents by inverse gas chromatography (IGC) is developed. In contrast with the methods described in the literature which are based on nonlinear chromatography (thermodynamics of adsorption) this method is based on linear chromatography (kinetics of adsorption). The mass balance equation of linear chromatography with axial diffusion term and the Langmuir kinetic equation on an open (non-porous) heterogeneous surface are solved by the method of Laplace transforms. The expressions for four semi-invariants of an elution profile are obtained. These are linear combinations of the moments of the distribution of sites in residence time of adsorbed molecule with coefficients determined from elution profile of non-adsorbing gas. Four semi-invariants of elution profiles of benzene and methane on a column packed with E-glass fiber were determined experimentally at temperatures around 100 degrees C. The mean, the standard deviation of the normal distribution in adsorption energy as well as the sticking coefficient and pre-exponential factor of the Frenkel equation are determined for benzene on E-glass. A method of the specific surface determination from linear IGC is proposed.     

A proposal for the estimation of binary mixture activity coefficients from surface tension measurements throughout the entire concentration range

A method to estimate the concentration dependence of the bulk activity coefficients of both binary mixture components from experimental surface tension data covering the whole concentration range is presented. To this end, high-quality experimental surface tension data as a function of the concentration, paying special attention to the diluted regions, are needed. The method is based on the application of equilibrium conditions to the surface chemical potential arisen from the Volmer equation (the simplest non-ideal surface EOS) and to the bulk chemical potential coming from the three-suffix Margules equation for activity coefficients. It can be applied provided that positive aneotropy is not present and the second derivative of the surface pressure as a function of the mole fraction is not positive in any composition region, which means rather unrestrictive conditions. In order to test the method, a compilation of 25 systems for which surface tension and activity coefficient data are simultaneously available in the literature was performed, turning out that most of them involved water. It was found that surface tension data with enough concentration coverage and good quality are really scarce, which makes the development of this kind of methods difficult.     

The surface properties of 1,3,5,7-tetranitro-1,3,5,7-tetraazacyclooctane after partial solution in nitro esters

The thermophysical properties and surface structure of 1,3,5,7-tetranitro-1,3,5,7-tetraazacyclooctane (TNCO) particles before and after their contact with liquid nitro esters were studied by differential scanning calorimetry and small-angle X-ray scattering. The thermophysical properties of TNCO changed after contact with nitro esters, and the temperature range of the transition from the β to δ polymorph broadened. This peculiarity was explained in terms of the fracton theory by changes in the heat capacity of particles related to the formation of nanoscale roughness with a fractal (scale-invariant) organization as a result of surface etching by solvents such as nitro esters, which are thermodynamically “bad” with respect to TNCO. The volume properties of particles did not change. The possibility of determining the fractal dimension of the surface of TNCO particles by differential scanning calorimetry was demonstrated. The rate of heating was then used as a “scale ruler” (“yardstick”), and the temperature range of the transition from the β to δ polymorph, as a measured parameter. The dimension of the surface fractal was also determined by an independent method (small-angle X-ray scattering). The results obtained by the two methods were closely similar.     

An Evaluation of Cellular Analyte-Receptor Binding Kinetics Utilizing Biosensors: A Fractal Analysis

A fractal analysis is presented for cellular analyte-receptor binding kinetics utilizing biosensors. Data taken from the literature can be modeled by using (a) a single-fractal analysis and (b) a single- and a dual-fractal analysis. Case (b) represents a change in the binding mechanism as the reaction progresses on the biosensor surface. Relationships are presented for the binding rate coefficient(s) as a function of the fractal dimension for the single-fractal analysis examples. In general, the binding rate coefficient is rather sensitive to the degree of heterogeneity that exists on the biosensor surface. For example, for the binding of mutagenized and back-mutagenized forms of peptide E1037 in solution to salivary agglutinin immobilized on a sensor chip, the order of dependence of the binding rate coefficient, k, on the fractal dimension, D(f), is 13.2. It is of interest to note that examples are presented where the binding coefficient (k) exhibits an increase as the fractal dimension (D(f)) or the degree of heterogeneity increases on the surface. The predictive relationships presented provide further physical insights into the binding reactions occurring on the surface. These should assist us in understanding the cellular binding reaction occurring on surfaces, even though the analysis presented is for the cases where the cellular "receptor" is actually immobilized on a biosensor or other surface. The analysis suggests possible modulations of cell surfaces in desired directions to help manipulate the binding rate coefficients (or affinities). In general, the technique presented is applicable for the most part to other reactions occurring on different types of biosensors or other surfaces. Copyright 2000 Academic Press.     

Analyte-receptor binding kinetics for different types of biosensors

A fractal analysis is presented for analyte-receptor binding kinetics for different types of biosensor applications. Data taken from the literature may be modeled using a single-fractal analysis, a single- and a dual-fractal analysis, or a dual-fractal analysis. The latter two methods represent a change in the binding mechanism as the reaction progresses on the surface. Predictive relationships developed for the binding rate coefficient as a function of the analyte concentration are of particular value since they provide a means by which the binding rate coefficients may be manipulated. Relationships are presented for the binding rate coefficients as a function of the fractal dimension D _f or the degree of heterogeneity that exists on the surface. When analyte-receptor binding is involved, an increase in the heterogeneity on the surface (increase in D _f ) leads to an increase in the binding rate coefficient. It is suggested that an increase in the degree of heterogeneity on the surface leads to an increase in the turbulence on the surface owing to the irregularities on the surface. This turbulence promotes mixing, minimizes diffusional limitations, and leads subsequently to an increase in the binding rate coefficient. The binding rate coefficient is rather sensitive to the degree of heterogeneity, D _f , that exists on the biosensor surface. For example, the order of dependence on D _f1 is 7.25 for the binding rate coefficient k ₁ for the binding of a Fab fragment of an antiparaquat monoclonal antibody in solution to an antigen in the form of a paraquat analog immobilized on a sensor surface. The predictive relationships presented for the binding rate coefficient and the fractal dimension as a function of the analyte concentration in solution provide further physical insights into the binding reactions on the surface, and should assist in enhancing biosensor performance. In general, the technique is applicable to other reactions occurring on different types of surfaces, such as cell-surface reactions.     

Prediction of three-dimensional fractal dimensions using the two-dimensional properties of fractal aggregates

Fractal dimension analysis using an optical imaging analysis technique is a powerful tool in obtaining morphological information of particulate aggregates formed in coagulation processes. However, as image analysis uses two-dimensional projected images of the aggregates, it is only applicable to one and two-dimensional fractal analyses. In this study, three-dimensional fractal dimensions are estimated from image analysis by characterizing relationships between three-dimensional fractal dimensions (D(3)) and one (D(1)) and two-dimensional fractal dimensions (D(2) and D(pf)). The characterization of these fractal dimensions were achieved by creating populations of aggregates based on the pre-defined radius of gyration while varying the number of primary particles in an aggregate and three-dimensional fractal dimensions. Approximately 2000 simulated aggregates were grouped into 33 populations based on the radius of gyration of each aggregate class. Each population included from 15 to 115 aggregates and the number of primary particles in an aggregate varied from 10 to 1000. Characterization of the fractal dimensions demonstrated that the one-dimensional fractal dimensions could not be used to estimate two- and three-dimensional fractal dimensions. However, two-dimensional fractal dimensions obtained statistically, well-characterized relationships with aggregates of a three-dimensional fractal characterization. Three-dimensional fractal dimensions obtained in this study were compared with previously published experimental values where both two-dimensional fractal and three-dimensional fractal data were given. In the case of inorganic aggregates, when experimentally obtained three-dimensional fractal dimensions were 1.75, 1.86, 1.83+/-0.07, 2.24+/-0.22, and 1.72+/-0.13, computed three-dimensional fractal dimensions using two-dimensional fractal dimensions were 1.75, 1.76, 1.77+/-0.04, 2.11+/-0.09, and 1.76+/-0.03, respectively. However, when primary particles were biological colloids, experimentally obtained three-dimensional fractal dimensions were 1.99+/-0.08 and 2.14+/-0.04, and computed values were both 1.79+/-0.08. Analysis of the three-dimensional fractal dimensions with the imaging analysis technique was comparable to the conventional methods of both light scattering and electrical sensing when primary particles are inorganic colloids.     

Fractal analysis of polyethylene catalysts surface morphologies based on wavelet transform modulus maxima method

Surface morphologies of supported polyethylene (PE) catalysts are investigated by an approach combining fractal with wavelet. The multiscale edge (detail) pictures of catalyst surface are extracted by wavelet transform modulus maxima (WTMM) method. And, the distribution of edge points on the edge image at every scale is studied with fractal and multifractal method. Furthermore, the singularity intensity distribution of edge points in the PE catalyst is analyzed by multifractal spectrum based on WTMM. The results reveal that the fractal dimension values and multifractal spectrums of edge images at small scales have a good relation with the activity and surface morphology of PE catalyst. Meanwhile the catalyst exhibiting the higher activity shows the wider singular strength span of multifractal spectrum based on WTMM, as well as the more edge points with the higher singular intensity. The research on catalyst surface morphology with hybrid fractal and wavelet method exerts the superiorities of wavelet and fractal theories and offers a thought for studying solid surfaces morphologies. Supported by the Chinese Petroleum & Chemical Corporation Development Department (Grant No. x504024)