Characterising latex particles and fractal aggregates using image analysis

The fractal nature of latex particles and their aggregates was characterised by image analysis in terms of fractal dimensions. The one- and two-dimensional fractal dimensions, D ₁ and D ₂, were estimated for polystyrene latex aggregates formed by flocculation in citric acid/phosphate buffer solutions. The dimensional analysis method was used, which is based on power law correlations between aggregate perimeter, projected area and maximum length. These aggregate characteristics were measured by image analysis. A two-slopes method using cumulative size distributions of aggregate length and solid volume has been developed to determine the three-dimensional fractal dimension (D ₃) for the latex aggregates. The fractal dimensions D ₁, D ₂ and D ₃ measured for single latex particles in distilled water agreed well with D ₁ = 1, D ₂ = 2 and D ₃ = 3 expected for Euclidean spherical objects. For the aggregates, the fractal dimension D ₂ of about 1.67 ± 0.04 (±standard deviation) was comparable to the fractal dimension D ₃ of approximately 1.72 ± 0.13 (±standard deviation), taking the standard deviations into account. The measured three-dimensional fractal dimension for latex aggregates is within the fractal dimension range 1.6–2.2 expected for aggregates formed through a cluster-cluster mechanism, and is close to the D ₃ value of about 1.8 indicated for cluster formation via diffusion-limited colloidal aggregation. Received: 28 September 1998 Accepted: 29 October 1998     

Fractal dimension and phase transition of graphene oxide (GO) doped polyacrylamide

Graphene Oxide (GO)- Polyacrylamide composites prepared between 5 and 50 μl GO were performed by Fluorescence Spectroscopy. The phase transition performed on the composites was measured by calculating the critical exponents, β and γ, respectively. In addition, fractal analysis of the composites was calculated by a fluorescence intensity of 427 nm. The geometrical distribution of GO in the composites was calculated based on the power law exponent values using scaling models. While the gelation proceeded GO plates first organized themselves into a 3D percolation cluster with the fractal dimension (D_f) of the composite, D_f = 2.63, then After it goes to diffusion limited clusters with D_f = 1.4, its dimension lines up to a Von Koch curve with a random interval of D_f = 1.14.     

Fractal Dimensions of Coals and Cokes

Using adsorption data, we get formulas for the calculation of fractal dimensions: log[A_CO2(DP)/A_N2(BET)] = −5.3984(2 −D₁)/2 and log[A_CO2(BET)/A_N2(BET)] = −4.9569(2 −D₂)/2. The fractal dimensions (D) of 27 coals and 2 cokes have been obtained. TheDof coals decreased with the increase of f_aand reached a maximum at H/C equal to 0.66 (orC_dafabout 86%). The fractal dimension is relative to ash and volatiles of coal:D= 2.2237 + 0.6249V_daf+ 0.8863A_d. The relationship betweenDof coal coke and its conversions (X) obeys the following equation:D = aexp(−bX) +c.     

Antigen-antibody binding kinetics for biosensors

The diffusion-limited binding kinetics of antigen in solution to antibody immobilized on a biosensor surface is analyzed within a fractal framework. Changes in the fractal dimension, D_f observed are in the same and in the reverse directions as the forward binding rate coefficientk. For example, an increase in the concentration of the isoenzyme human creatine kinase isoenzyme MB form (CK-MB) (antigen) solution from 0.1 to 50 ng/mL and bound to anti-CK-MB antibody immobilized on fused silica fiber rods leads to increases in the fractal dimension D_f from 0.294 to 0.5080, and in the forward binding rate coefficientk from 0.1194 to 9.716, respectively. The error in the fractal dimension D_f decreases with an increase in the CK-MB isoenzyme concentration in solution. An increase in the concentration of human chorionic gonadotrophin (hCG) in solution from 4000 to 6000 mIU/mL hCG and bound to anti-hCG antibody immobilized on a fluorescence capillary fill device leads to a decrease in the fractal dimension D_f from 2.6806 to 2.6164, and to an increase in the forward binding rate coefficientk from 3.571 to 4.033, respectively. The different examples analyzed and presented together indicate one means by which the forward binding rate coefficientk may be controlled, that is by changing the fractal dimension or the ‘disorder’ on the surface. The analysis should assist in helping to improve the stability, the sensitivity, and the response time of biosensors.     

Analyte-receptor binding kinetics for biosensor applications

The diffusion-limited binding kinetics of antigen (analyte), in solution with antibody (receptor) immobilized on a biosensor surface, is analyzed within a fractal framework. Most of the data presented is adequately described by a single-fractal analysis. This was indicated by the regression analysis provided by Sigmaplot. A single example of a dual-fractal analysis is also presented. It is of interest to note that the binding-rate coefficient (k) and the fractal dimension (D_f) both exhibit changes in the same and in the reverse direction for the antigen-antibody systems analyzed. Binding-rate coefficient expressions, as a function of the D_f developed for the antigen-antibody binding systems, indicate the high sensitivity of thek on the D_f when both a single- and a dual-fractal analysis are used. For example, for a single-fractal analysis, and for the binding of antibody Mab 0.5β in solution to gpl20 peptide immobilized on a BIAcore biosensor, the order of dependence on the D_f was 4.0926. For a dual-fractal analysis, and for the binding of 25-100 ng/mL TRITC-LPS (lipopolysaccharide) in solution with polymyxin B immobilized on a fiberoptic biosensor, the order of dependence of the binding-rate coefficients, k₁ and k₂ on the fractal dimensions, D_f1 and D_f2, were 7.6335 and-11.55, respectively. The fractional order of dependence of thek(s) on the D_f(s) further reinforces the fractal nature of the system. Thek(s) expressions developed as a function of the D_f(s) are of particular value, since they provide a means to better control biosensor performance, by linking it to the heterogeneity on the surface, and further emphasize, in a quantitative sense, the importance of the nature of the surface in biosensor performance.     

Growth process for fractal polymer aggregates formed by perfluorooctyltrimethoxysilane. Time-resolved small-angle X-ray scattering study

The acid-catalyzed condensation reaction of perfluorooctyltrimethoxysilane (PFOS) and n-octyltrimethoxysilane (OTMS) in ethanol has been followed by time-resolved synchrotron radiation small-angle X-ray scattering (SAXS) on a short time scale. SAXS curves for PFOS and OTMS have been interpreted as arising from mass fractals with D _f=2 (PFOS) and D _f=1.7 (OTMS). The time dependence of the apparent radius of gyration, obtained from the Guinier plots, showed that the growth of fractal precursors occurs in a two-step process within 2 h for PFOS and within 1.5 h for OTMS, in which small clusters involving monomers, dimers and trimers are formed in the initial step and formation of larger clusters occurs in the second step. Furthermore, it has been suggested that the hydrophobicity and lipophobicity of the bulky alkyl groups may also contribute to the formation of these giant aggregates. Received: 13 July 1999/Accepted in revised form: 6 October 1999     

Viscoelastic scaling in polymer gels

New scaling laws for chain networks are derived to describe the fundamental relationships between the viscosity exponent (k), viscoelastic exponent (m), stretched exponent (β), spatial dimension (d). fractal dimension (d_f), and a universal constant (γ). The scaling of the total number of monomers and the radius of gyration is defined by d_f. We have discovered γ = m/β to be a universal constant which relates the shear modulus of a polymer gel melt to the shear modulus near the glass transition. Analyzing the size-dependent shear viscosity, we have determined γ = 3d_fcd/(7d−5d_fc) = 2.647 for d = 3 where d_fc is the fractal dimension of critical clusters at the gel point. By using γ, the present theory extends previous work pertaining to systems near the sol-gel transition, and shows how properties far from the critical point can be explained. The theoretical prediction is in good agreement with viscoelastic measurements.     

Size exclusion chromatography on porous fractals

Summary Some porous packings used in chromatography have been claimed to be fractals with a scale of sizes a<l<L, where a is a molecular size and L is the size of the largest pores. For a fractal porous packing, the excluded volume for molecules in solution in the vicinity of the packing surface is directly related to D_f, the fractal dimension of the pore surface (2<D_f<3). Since retention in size exclusion chromatography is itself directly related to this excluded volume, the fractal nature of the packing provides a model of retention in this technique. According to this model there is a linear relationship between log R_s and log(1-K_d), where R_s is the hydrodynamic radius of the solute macromolecules and K_d the distribution coefficient. The fractal dimension is derived from the slope of this plot. Size exclusion chromatographic retention data have been analyzed according to the model. It is found that some HPLC packings are fractals with fractal dimensions ranging from about 2.15 to 2.6, depending on the material. Such a large range of D_f values indicates large variations in the selectivities and domains of applications of the different packings. For some classical gel filtration chromatographic gels, the fractal retention model does not seem to apply.Presented at the 17th International Symposium on Chromatography, September 25–30, 1988, Vienna, Austria.     

Studies of physico-chemical properties of mixed adsorbent in the zeolite/SiO2 system

The paper presents physico-chemical properties of mixed adsorbents in the clinoptylolite (mordenite)/SiO₂ system containing 30, 50, 80 mass% zeolite. Adsorption capacity towards polar (water, butanol) and non-polar (n-octane) substances as well as total surface heterogeneity (energetic and geometrical) were determined. Desorption energy distribution functions as well as fractal dimensions were also determined and compared with the low-temperature nitrogen adsorption data. Irregular shapes of the curves q=f(E _d) as well as large values of volumetric fractal dimensions (D _f~2.6) revealed heterogeneous properties of the zeolite/SiO2 system surfaces. Addition of zeolite increases total heterogeneity of the material.     

Monitoring coalescence behavior of soft colloidal particles in water by small-angle light scattering

The fractal dimension (D _f) of the clusters formed during the aggregation of colloidal systems reflects correctly the coalescence extent among the particles (Gauer et al., Macromolecules 42:9103, 2009). In this work, we propose to use the fast small-angle light scattering (SALS) technique to determine the D _f value during the aggregation. It is found that in the diffusion-limited aggregation regime, the D _f value can be correctly determined from both the power law regime of the average structure factor of the clusters and the scaling of the zero angle intensity versus the average radius of gyration. The obtained D _f value is equal to that estimated from the technique proposed in the above work, based on dynamic light scattering (DLS). In the reaction-limited aggregation (RLCA) regime, due to contamination of small clusters and primary particles, the power law regime of the average structure factor cannot be properly defined for the D _f estimation. However, the scaling of the zero angle intensity versus the average radius of gyration is still well defined, thus allowing one to estimate the D _f value, i.e., the coalescence extent. Therefore, when the DLS-based technique cannot be applied in the RLCA regime, one can apply the SALS technique to monitor the coalescence extent. Applicability and reliability of the technique have been assessed by applying it to an acrylate copolymer colloid.     

Disruption of polystyrene latex aggregates in capillary flow

 Disruption of polystyrene latex aggregates, formed in 1 M citric acid/phosphate buffer solution at pH 3.8 through diffusion-limited colloid aggregation (DLCA) and in 0.2 M NaCl solution at pH 5.5 through reaction-limited colloid aggregation (RLCA), was studied with respect to aggregate size and fractal nature. This was achieved using small-angle laser scattering in conjunction with a specially designed sampling method, which brought about the elimination of the disruption of the aggregates caused by a commercial stirrer sample unit. Aggregations were carried out in a mixture of deuterium oxide and water instead of water alone as a solvent to minimise sedimentation resulting from the differences in density between the latex particles and the electrolytes. An initial “steady state” in terms of aggregate size and fractal dimension was found to occur after around 20 min and 2 days for DLCA and RLCA aggregates, respectively, at 25 °C. No aggregate disruption was detected for DLCA and RLCA aggregates after their passing through a capillary tube for shear rates up to 1584 and 2694 s⁻¹, respectively. At higher shear rates, significant decreases in the aggregate volume-mean diameter, D[4, 3], occurred after shearing. The degree of reduction in D[4, 3] was larger for DLCA aggregates in comparison to RLCA aggregates. The results would suggest that DLCA aggregates were more subject to disruption during shearing. A high degree of disruption was observed in turbulent flow for both aggregates. Received: 30 June 1999 Accepted in revised form: 11 November 1999     

STEREOELECTRONIC PROPERTIES OF PHOTOSYNTHETIC AND RELATED SYSTEMS—VIII. AB INITIO QUANTUM MECHANICAL CHARACTERIZATION OF THE ELECTRONIC STRUCTURE AND SPECTRUM OF THE BACTERIOPHEOPHORBIDE a ANION RADICAL

Ab initio configuration interaction wavefunctions and energies are reported for 16 doublet states of the anion radical of ethyl bacteriopheophorbide a (Et-BPheo a^-˙), and are employed in an analysis of the electronic absorption spectrum. The lowest excited doublet state D₁ is predicted to lie 8601 cm^-1 above the ground state D₀; the D₁← D₀ transition is nearly forbidden, with a computed oscillator strength f= 0.002. The visible absorption spectrum is shown to consist of transitions to three ²(π, π*) states, D₂, D₃, and D₄. The D₄← D₀ transition (y-polarized, f= 0.91) appears to account for observed intense absorption at 15 800 cm^-1. The Soret band of Et-BPheo a^-˙ is shown to consist of transitions to several ²(π,π*) states, D₇-D₁₅. Transitions of particularly high intensity include D₇← D₀ (y-polarized, f= 0.72), D₁₀← D₀ (y-polarized, f= 1.1), D₁₂← D₀ (xy-polarized, f= 0.86) and D₁₅← D₀ (y-polarized, f= 0.83). Spin density data and plots are used to describe and compare the general features of the unpaired spin distributions in D₀ and D₁, which are in reasonable agreement with other reported calculated values and available experimental data for D₀.     

Computer Simulation of Bridging Flocculation Processes: The Role of Colloid to Polymer Concentration Ratio on Aggregation Kinetics

The flocculation of colloidal particles by adsorbing polymers is one of the central issues of colloid science and a very important topic in many industrial, biological, and environmental processes. We report a computer simulation study of a 2- and 3-dimensional model for bridging flocculation betweenlarge linear polymer chains and comparatively small colloidal particles,where the structure and growth kinetics of cluster formation are investigated. This model was developed within the framework of the cluster–cluster aggregation model using mass and fractal dimension dependent diffusion constants, where bridging flocculation is seen as a case of heterocoagulation in which, in addition, macromolecule configurations and lengths play an important role. The simulation of aggregate structure and formation kinetics obtained at different (i) relative particle concentrations, (ii) polymer chain conformations, and (iii) sticking probabilities are described from a qualitatively and quantitative point of view. The results suggest that the formation of large aggregates is a slow process, controlled by the reactivity of the clusters, even when the reaction between microcolloids and macrochains is very fast. Aggregation kinetics are strongly dependent on the particle/chain concentration ratio and on the configurational properties of the chains. It is shown that the scaling laws which are valid for homocoagulation processes are also applicable to the kinetics of bridging flocculation. The corresponding scaling exponents have been calculated.     

Thermoanalytical study of cyclization reaction of some thioureates

Thermal initiated conversion of N-aryl-N'-(2-benzylpyridinium)thioureates into 2-arylamino-4H-benzo[d][1,3]thiazines was studied by non-isothermal differential scanning calorimetry (DSC), thermogravimetry (TG) and differential thermal analysis (DTA) in the solid-state. The values of molar reaction ethalpies (DH _r) of six derivatives of thioureates and the melting parameters (T _f, DH _f, DS _f) of the obtained products - benzothiazines were determined by the DSC method. This revised version was published online in August 2006 with corrections to the Cover Date.     

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.     

Theoretical and experimental analyses of the deposited silver thin films

The silver thin films have been prepared using magnetron DC‐sputtering. We discuss in detail the thin films AFM images and their properties in different sputtering times of 2 to 6 minutes. Despite the low thickness of the films, the roughness saturation amounts, W_s, are well separated. The surface data do not follow the normal Family‐Vicsek scaling, and we have the local growth exponent, β(W_s(t)∼t^β). We obtained the global roughness scaling exponent α=0.36 and growth exponent, β=0.50. We also obtain the fractal spectrum of the data, f(α). The results show that the spectrum is right‐hook like. It distinguishes between different film thicknesses even in small sizes of hundreds of nanometers. Furthermore, we measure the surface conductivities and compare them to the thin film roughnesses. We investigate the roughness and fractality of the AFM data, looking for their relations to width and conductivity of the silver thin film samples.     

Characterization of Titania/Silica Gel by Means of Low-Pressure Nitrogen Adsorption

Adsorbents synthesized by grafting of titania onto mesoporous silica gel surfaces at different temperatures were studied by means of nitrogen adsorption–desorption and water desorption. The pore size distribution f(R_p) of titania/silica gel depends on the titania concentration (C_TiO2) and the temperature of titania synthesis. Nonuniformity of TiO₂ phase is maximal at a low C_TiO2 value (3.2 wt.% anatase deposited at 473 K), and two peaks of the fractal dimension distribution f(D) are observed at such a concentration of titania, but at larger C_TiO2 values, only one f(D) peak is seen. More ordered filling of pores and adsorption sites by nitrogen, reflecting in the shape of adsorption energy distributions f(E) at different pressures of adsorbate, is observed for adsorbent with titania (rutile+anatase) grafted on silica gel at a higher temperature (673 K).     

The origin of anomalous enhancement of electromagnetic fields in fractal aggregates of metal nanoparticles

It is shown that the anisotropy of the environment of metal nanoparticles with plasmon absorption in fractal aggregates is the most important and universal characteristic underlying their unique electrodynamic properties. It is noted that it is this morphological feature, but not the fractal distribution of particles in aggregates as such, that plays the dominant role in the manifestation of the enhancement of a local field. In this case, fractal aggregates possess the ability to enhance local electromagnetic fields only owing to their inherent local anisotropy; macroscopic characteristics of aggregates do not markedly influence their electrodynamic interactions with the external field. The quantitative characteristic of local anisotropy is introduced. Statistical correlation between the factor of local anisotropy and fractal dimension D of aggregates is established within the range 1.6 < D < 2.8. It is disclosed that the local anisotropy is independent of the fractal dimension within the wide range (1.6 < D < 2.5) except for the range D > 2.5 corresponding to aggregates with close-packed particles where the factor of local anisotropy tends toward zero. Strong correlation in the spatial arrangement of particles with the largest local anisotropy of the environment in aggregates and the strength of the local electromagnetic field is established using aggregates of silver nanoparticles as an example for the spectrum in the visible range; a polarization dependence of this correlation is revealed. It is noted that parameters of local anisotropy can be used to determine the degree of imperfection of colloidal crystals via optical methods.     

Diffusion coefficients of the monomer and oligomers in hydroxyethyl methacrylate

The diffusion coefficients are reported of rubbery ternary systems consisting of the polymer, its monomer analogue (i.e., the saturated equivalent of the monomer), and trace quantities of oligomers (dimer, trimer, tetramer and hexamer) for 2‐hydroxyethyl methacrylate (HEMA). These have been obtained with pulsed‐field‐gradient NMR spectroscopy with a polymer weight fraction (f_p) of 0 ≤ f_p ≤ 0.4. The oligomers are macromonomers synthesized with a cobalt catalytic chain‐transfer agent. The diffusion coefficients are about an order of magnitude smaller than those for monomers such as methyl methacrylate; this effect is ascribed to hydrogen bonding in HEMA. The diffusion coefficient D_i of an i‐meric oligomer has been fitted with moderate accuracy by an empirical universal scaling relation, D_i(f_p)/D₁(f_p) ≈ i, previously found to provide an adequate fit to corresponding data for styrene and for methyl and butyl methacrylates. The approximate empirical scaling relation seems to hold for a remarkably wide range of types of monomer/polymer systems. These results are of use in modeling rates and molecular weight distributions in free‐radical polymerization, particularly for termination (which is chain‐length‐dependent and is controlled by the diffusion coefficient of chains of the low degrees of polymerization studied here). © 2003 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 41: 2491–2501, 2003