Low Gel Concentrations Research Articles

Phenanthrothiophene-Triazine Star-Shaped Discotic Liquid Crystals: Synthesis, Self-Assembly, and Stimuli-Responsive Fluorescence Properties.

Lipophilic biphenylthiophene- and phenanthrothiophene-triazine compounds, BPTTn and CPTTn, respectively, were prepared by a tandem procedure involving successive Suzuki-Miyaura coupling and Scholl cyclodehydrogenation reactions. These compounds display photoluminescence in solution and in thin film state, solvatochromism with increasing solvent's polarity, as well as acidochromism and metal ion recognition stimuli-responsive fluorescence. Protonation of BPTT10 and CPTT10 by trifluoroacetic acid results in fluorescence quenching, which is reversibly restored once treated with triethylamine (ON-OFF switch). DFT computational studies show that intramolecular charge transfer (ICT) phenomena occurs for both molecules, and reveal that protonation enhances the electron-withdrawing ability of the triazine core and reduces the band gap. This acidochromic behavior was applied to a prototype fluorescent anti-counterfeiting device. They also specifically recognize Fe3+ through coordination, and the recognition mechanism is closely related to the photoinduced electron transfer between Fe3+ and BPTT10/CPTT10. CPTTn self-assemble into columnar rectangular (Colrec) mesophase, which can be modulated by oleic acid via the formation of a hydrogen-bonded supramolecular liquid crystal hexagonal Colhex mesophase. Finally, CPTTn also form organic gels in alkanes at low critical gel concentration (3.0 mg/mL). Therefore, these star-shaped triazine molecules possess many interesting features and thus hold great promises for information processing, liquid crystal semiconductors and organogelators.

PH-Sensitive Poly(acrylic acid)-g-poly(L-lysine) Charge-Driven Self-Assembling Hydrogels with 3D-Printability and Self-Healing Properties.

We report the rheological behavior of aqueous solutions of a graft copolymer polyampholyte, constituted of polyacrylic acid (PAA) backbone grafted by Poly(L-lysine) (PAA-b-PLL). The graft copolymer self-assembles in aqueous media, forming a three-dimensional (3D) network through polyelectrolyte complexation of the oppositely charged PAA and PLL segments. Rheological investigations showed that the hydrogel exhibits interesting properties, namely, relatively low critical gel concentration, elastic response with slow dynamics, remarkable extended critical strain to flow, shear responsiveness, injectability, 3D printability and self-healing. Due to the weak nature of the involved polyelectrolyte segments, the hydrogel properties display pH-dependency, and they are affected by the presence of salt. Especially upon varying pH, the PLL secondary structure changes from random coil to α-helix, affecting the crosslinking structural mode and, in turn, the overall network structure as reflected in the rheological properties. Thanks to the biocompatibility of the copolymer constituents and the biodegradability of PLL, the designed gelator seems to exhibit potential for bioapplications.

Effect of targeting ligand designation of self-assembly chitosan-poloxamer nanogels loaded Paclitacel on inhibiting MCF-7 cancer cell growth

In this study, we investigated two formulations of chitosan-Pluronic P123 with different folate ligand designation for targeted delivery of Paclitaxel (PTX), in which folic acid (FA) was directly conjugated to chitosan (FA-Cs-P123) or substituted onto P123 (Cs-P123-FA). The results showed that the FA content of Cs-P123-FA was determined at 0.71 wt/wt% which was significantly higher than that of FA-Cs-P123 (0.31 wt/wt%). Two copolymers were low critical gel concentrations (CGC). FA-Cs-P123 and Cs-P123-FA nanogels performed high PTX encapsulation efficiency reaching 95.57 ± 5.51 and 92.51 ± 6.68 wt/wt%, respectively. Transmission electron microscopy (TEM) and zeta potential analysis indicated that the PTX-loaded nanogels were spherically formed around 60 nm in diameter along with positive charge. Furthermore, the PTX release profile was slow and it was controlled by the pH of the medium. In particular, in vitro biocompatibility assays indicated that both FA-Cs-P123 and Cs-P123-FA exhibited good biological compatibility with a human foreskin fibroblast cell line and well uptake efficiency into MCF-7 cancer cells. Cs-P123-FA nanogel significantly enhanced the cytotoxicity of PTX in comparison with FA-Cs-P123. The result indicates that Cs-P123-FA nanogels with a higher decorated FA content perform a better targeting efficiency; therefore, they could have great potential application towards breast cancer treatment.

Printability and bio-functionality of a shear thinning methacrylated xanthan–gelatin composite bioink

3D bioprinting is a recent technique that can create complex cell seeded scaffolds and therefore holds great promise to revolutionize the biomedical sector by combining materials and structures that more closely mimic the 3D cell environment in tissues. The most commonly used biomaterials for printing are hydrogels, however, many of the hydrogels used still present issues of printability, stability, or poor cell-material interactions. We propose that bioinks with intrinsic self-assembling and shear thinning properties, such as xanthan gum, can be methacrylated (XGMA) and combined with a bio-functional material such as gelatin methacryloyl (GelMa) to create a stable, cell-interactive bioink with improved properties for 3D bioprinting. These biomaterials have reduced viscosity under high shear and recover their viscosity rapidly after the shear is removed, retaining their shape, which translates to easier extrusion whilst maintaining accurate fidelity after printing. This was confirmed in printing studies, with measured normalized strand widths of 1.2 obtained for high gel concentrations (5+5 % XGMA-GelMA). Furthermore, the introduction of a secondary photo-cross-linking method allowed tuning of the mechanical properties of the hydrogel with stiffness between 15 and 30 kPa, as well as improving the stability of the hydrogel with retention of 75 % of its mass after 90 d. The hydrogel was shown to be biocompatible and bio-active with 97 % cell viability, and cell spreading after 7 d of culture for low gel concentrations (3+3 % XGMA-GelMA). Shear stresses were relatively low while printing (1 kPa) as a result of the shear thinning property of the material, which supported cell viability during extrusion. Finally, printed hydrogels retained high cell viability for lower gel concentrations, and showed improved cell viability for more concentrated hydrogels when compared to cells cultured in bulk hydrogels, presumably due to improved nutrient/oxygen diffusion and cell migration. In conclusion, stability and formulation of a XGMA-GelMA shear thinning composite hydrogel has been optimized to create a bio-functional bioink, with improved printability, and in vitro culture stability via secondary photo-induced cross-linking, making this composite a promising bioink for 3D bioprinting.

An in-depth study on the agar gel effectiveness for built heritage cleaning

The cleaning effectiveness of agar gels for copper stain removal from marble is mainly related to the gel morphology and to the possibility of copper coordination. • Different agar gel formulations were tested for copper stain removal from marble. • Hydrogel lyophilisation allowed to systematically compare gel formulations. • The relationship between gel morphology and effectiveness was assessed. • AgarArt 1% with TAC was the most effective gel for copper removal from marble. • EDTA in gels leads to a broader pore size distribution and to a lower gel strength. The effectiveness of Agar gels for copper stain removal from marble surfaces was systematically studied. Gels with different agar concentrations (1, 3, 5%) and different chelating agents used as additives (ethylenediaminetetraacetic acid, EDTA, and ammonium citrate tribasic, TAC) were tested on laboratory marble specimens for different contact times (30 and 60 min). For better characterization, hydrogels were lyophilised and cryogels were obtained. Systematic comparison of different formulations was feasible on cryogels and performed in terms of: (i) the morphological properties, by field-emission scanning electron microscopy (FE-SEM); (ii) the type of Cu(II)-complexes formed and their quantitative comparison by electron paramagnetic resonance (EPR) spectroscopy; (iii) the total amount of copper removed from marble surfaces, by Inductively Coupled Plasma–Optical Emission Spectrometry (ICP-OES). AgarArt 1% with TAC exhibited the highest effectiveness for copper stain removal after 60 min contact (431 μg/cm 2 ). Such a good cleaning effectiveness can be ascribed to the co-presence of the following properties: efficient metal coordination, which is related to the additive presence, and favourable gel morphology, related both to the gel concentration and to the additive type. In fact, it was observed that both the low gel concentration and the presence of TAC are related to a narrow pore size distribution in gels, besides the possibility of copper coordination. The presence of EDTA results in a broader pore size distribution and in a lower gel strength, with respect to gels with TAC. Thus, a new procedure for studying gels was proposed, which allows to optimize the conditions for metal stain removal from built heritage.

Gel-Modulated Growth of High-Quality Zeolite Membranes

Trade-off between thickness (and thus gas permeance) and quality (and thus selectivity) of zeolite membranes significantly restricts their wide application. It is challenging to maintain the membrane thickness while minimizing nonselective defects in the selective membrane layer. Currently, continuous change of the synthesis gel concentration during membrane synthesis, instead of high gel concentration for membrane "skeleton" growth and low gel concentration for slow crystal growth to merge "skeleton" crystals, usually leads to thick membranes to compensate for the low selectivity. In this work, we report a gel-modulated synthesis approach to engineer the zeolite membrane synthesis. The gel concentration was suddenly reduced by a quenching process in the middle of the membrane synthesis, leading to reduced nuclei formation and crystal growth rate in the following synthesis process. Membrane quality was significantly improved without increasing membrane thickness, leading to a great increase in membrane selectivity but without sacrifice of gas permeance. Moreover, as an example, highly reproducible SAPO-34 membranes were successfully prepared by the gel-modulated growth method. We expect this novel synthesis strategy might be a viable and economic way of growing thin and high-quality zeolite membranes.

Biomimetic Design of Oxidized Bacterial Cellulose-gelatin-hydroxyapatite Nanocomposites

Oxidized Bacterial Cellulose (OBC)-hydroxyapatite (HAp)-gelatin (Gel) nanocomposites were prepared by a biomimetic process. HAp nanocrystals were precipitated in a mixed solution of Na2HPO4 (pH 9.2) and Gel solution at 37 °C, and OBC was used to generate a three-dimensional (3D) network stent. The tensile strength of OBC-HAp-G was higher than 0.3 MPa, and the complete degradation time was approximately 90 d in Simulated Body Fluid (SBF). Fourier transform infrared spectroscopy demonstrated that a coordinate bond had formed possibly between HAp and the cellulose hydroxyl. X-ray diffraction showed that both the oxidation of bacterial cellulose and an increase in Gel content induced the formation of tiny HAp crystallites during composite fabrication. Specific surface area and porosity measurements indicated that a low Gel concentration contributed to retention of porous structure. The Ca and P contents on the surface of materials increased initially and then decreased with an increase in Gel content, as measured by energy dispersive spectroscopy. From the thermogravimetric data, the increase in decomposition temperature suggested the formation of chemical bonds among OBC, HAp, and Gel. The above results suggest that the OBC-HAp-G0.3 composite is a potential bone scaffold material.

The impact of damage accumulation on the kinetics of network strength recovery for a physical polymer gel subjected to shear deformation

ABSTRACTShear rheophysical experiments were used to quantify the kinetics of strength recovery of model thermoreversible polymer gels that were fractured and ultimately sheared to different total magnitudes of strain (700 and 4000%) before resting for set periods of time. Relationships between the amount of strength recovered and the normalized ratio of resting times to characteristic relaxation times were developed. It was found that gels displayed fully healed networks within timescales that were 2–3 orders of magnitude greater than the gel's characteristic relaxation time. Gels deformed to 700% applied strain either healed slower at lower gel concentrations as compared to experiments at larger applied strains due to possible viscous heating, or healed faster from incomplete fracture propagation for higher gel concentrations. © 2016 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2016, 54, 1693–1701

Soft hydrogels interpenetrating silicone--A polymer network for drug-releasing medical devices.

Materials for the next generation of medical devices will require not only the mechanical stability of current devices, but must also possess other properties such as sustained release of drugs in a controlled manner over a prolonged period of time. This work focuses on creating such a sophisticated material by forming an interpenetrating polymer network (IPN) material through modification of silicone elastomers with a poly(2-hydroxyethyl methacrylate) (PHEMA)-based hydrogel. IPN materials with a PHEMA content in the range of 13%-38% (w/w) were synthesized by using carbon dioxide-based solvent mixtures under high pressure. These IPNs were characterized with regard to microstructure as well as ability of the hydrogel to form a surface-connected hydrophilic carrier network inside the silicone. A critical limit for hydrogel connectivity was found both via simulation and by visualization of water uptake in approximately 25% (w/w) PHEMA, indicating that entrapment of gel occurs at low gel concentrations. The optimized IPN material was loaded with the antibiotic ciprofloxacin, and the resulting drug release was shown to inhibit bacterial growth when placed on agar, thus demonstrating the potential of this IPN material for future applications in drug-releasing medical devices.

Emergence of Large Chiroptical Responses by Ligand Exchange Cross-Linking of Monolayer-Protected Gold Clusters with Chiral Dithiol

We here present a study of cross-linking chemistry of optically inactive monothiol-protected gold clusters by chiral bidentate dithiol with two stereogenic centers, (2R,3R)-1,4-dimercapto-2,3-butanediol (L-dithiothreitol; L-DTT), and explore the impacts of the cross-linking on their chiroptical responses. The pristine protective ligand is racemic penicillamine (rac-Pen), and the products of the ligand exchange reactions include clusters containing both rac-Pen and L-DTT (partial exchange). Electrophoresis using polyacrylamide gel with a very low gel concentration (3%) can make the products separable into two components, each of which has the similar mean core diameter of 0.78 and 0.83 nm, so the difference in the relative mobility is mainly ascribed to the size of the cluster assembly. In addition, very large optical activity with the maximum anisotropy factors of about 1.0 × 10(-3) is found for the assemblies. In comparison with chiral 1,3-dithiol protection incapable of cross-linking between gold clusters, we propose that the observed optical activity is due to surface intrinsic handedness caused by a cyclic cross-linking with at least two L-DTT molecules.

Oligo(p‐phenylene‐ethynylene)‐Derived Super‐π‐Gelators with Tunable Emission and Self‐Assembled Polymorphic Structures

Linear π-conjugated oligomers are known to form organogels through noncovalent interactions. Herein, we report the effect of π-repeat units on the gelation and morphological properties of three different oligo(p-phenylene-ethynylene)s: OPE3, OPE5, and OPE7. All of these molecules form fluorescent gels in nonpolar solvents at low critical gel concentrations, thereby resulting in a blue gel for OPE3, a green gel for OPE5, and a greenish yellow gel for OPE7. The molecule-molecule and molecule-substrate interactions in these OPEs are strongly influenced by the conjugation length of the molecules. Silicon wafer suppresses substrate-molecule interactions whereas a mica surface facilitates such interactions. At lower concentrations, OPE3 formed vesicular assemblies and OPE5 gave entangled fibers, whereas OPE7 resulted in spiral assemblies on a mica surface. At higher concentrations, OPE3 and OPE5 resulted in super-bundles of fibers and flowerlike short-fiber agglomerates when different conditions were applied. The number of polymorphic structures increases on increasing the conjugation length, as seen in the case of OPE7 with n=5, which resulted in a variety of exotic structures, the formation of which could be controlled by varying the substrate, concentration, and humidity.

Colloidal crystallization of thermo-sensitive gel spheres of poly (N-isopropyl acrylamide). Influence of degree of cross-linking of the gels

Morphology, phase diagram, and reflection spectroscopy of the colloidal crystals of thermo-sensitive gel spheres, poly (N-isopropylacrylamide) having degrees of cross-linking 10 and 2 mol.% (pNIPAm(200–10) and pNIPAm(200–2)) were studied. Giant colloidal single crystals formed at very low gel concentrations. Critical concentrations of melting increased as the degree of cross-linking decreased in the range from 10 to 0.5 mol.% and/or suspension temperature increased from 20 to 45 °C. The critical concentration decreased sharply as the suspensions were deionized with coexistence of the mixtures of cation- and anion-exchange resins. Density of a gel sphere (gel concentration in weight percent divided by that in volume percent) increased sharply as the degree of cross-linking and/or temperature increased. These results demonstrated that the colloidal crystallization takes place by the extended electrical double layers formed around the gel spheres in addition of the excluded-volume effect of the gels. Most of the researchers including the authors have believed that the crystallization of the gel spheres takes place by the excluded-volume effect. However, the present work clarified that the colloidal interfaces, which are inevitable for the formation of the electrical double layers, are formed firmly between the water phase and gel spheres, though the gel spheres contain a lot of water molecules in the sphere region.

Colloidal crystallization of thermosensitive gel spheres of poly(N-isopropylacrylamide) with low degree of cross-linking

Morphology, phase diagram, and reflection spectroscopy of the colloidal crystals of thermosensitive gel spheres, poly(N-isopropylacrylamide) ((200–0.5), 318 and 116 nm in the hydrodynamic diameter at 25°C and 45°C, and 0.5% in the degree of cross-linking) were studied. Giant colloidal single crystals formed at very low gel concentrations. Densities of the gel spheres were 0.030 and 0.61 at 25°C and 45°C, respectively. Critical concentration of melting of gel spheres (0.8 wt.% without ion-exchange resins) decreased sharply to 0.015 wt.% at 25°C as the gel suspension was deionized exhaustively with coexistence of the mixtures of cation and anion exchange resins. These results demonstrate that the colloidal crystallization takes place by the extended electrical double layers formed around the gel spheres in addition of the excluded volume effect of the gels. Extent of the contribution of the electrical double layers on the crystallization increased sharply when the degree of cross-linking increased, the gel spheres shrank, and/or the density of the gel spheres increased.

Rotational Diffusion of Macromolecules and Nanoparticles Modeled as Non-Overlapping Bead Arrays in an Effective Medium

In this work, the retarding influence of a gel on the rotational motion of a macromolecule is investigated within the framework of the Effective Medium (EM) model. This is an extension of an earlier study that considered the effect of a gel on the translational motion of a macromolecule [Allison, S. et al. J. Phys. Chem. B 2008, 112, 5858-5866]. The macromolecule is modeled as an array of non-overlapping spherical beads with no restriction placed on their size or configuration. Specific applications include the rotational motion of right circular cylinders and wormlike chains modeled as strings of identical touching beads. The procedure is then used to examine the electric birefringence decay of a 622 base pair DNA fragment in an agarose gel. At low gel concentration (M ≤ 0.010 gm/mL), good agreement between theory and experiment is achieved if the persistence length of DNA is taken to be 65 nm and the gel fiber radius of agarose is taken to be 2.5 nm. At higher gel concentrations, the EM model substantially underestimates the rotational relaxation time of DNA and this can be attributed to the onset of direct interactions that become significant when the effective particle size becomes comparable to the mean gel fiber spacing.

Colloidal crystallization of thermo-sensitive gel spheres of poly (N-isopropyl acrylamide)

Morphology, phase diagram, and reflection spectroscopy of the colloidal crystals of thermo-sensitive gel spheres, poly (N-isopropylacrylamide) (224 nm in the hydrodynamic diameter at 25 °C) were studied. Giant colloidal single crystals formed at very low gel concentrations. Critical concentration of melting of gel spheres (0.8 wt.% without ion-exchange resins) decreased sharply to 0.01 wt.% as the gel suspension was deionized exhaustively with coexistence of the mixtures of cation- and anion-exchange resins and increased substantially as concentration of sodium chloride increased. These studies demonstrated that the colloidal crystallization takes place by the extended electrical double layers formed around the gel spheres in addition of the excluded-volume effect of the gels. Most of the researchers including the authors have believed that the crystallization of the gel spheres takes place by the excluded-volume effect, in other words, by the hard-sphere model, exclusively. However, the present work clarified that the colloidal interfaces, which are inevitable for the formation of the electrical double layers, are formed firmly between the water phase and gel spheres, though the gel spheres contain a lot of water molecules in the inner the sphere region.

Mineralization of Calcium Carbonates in Gelling Media

Precipitation of calcium carbonate in different carrageenan gelling environments was initiated by fast mixing of calcium chloride and sodium carbonate solutions. Carrageenan concentration and the gelling mechanism, controlled by the addition of different cross-linking cations, calcium or sodium, were varied in order to elucidate the mechanism and dynamic of solid phase formation in highly supersaturated precipitation systems in which initial formation of three calcium carbonate polymorphs and amorphous calcium carbonate is possible. The analysis of the samples isolated 5 days after initiating the precipitation showed that calcite was the only solid phase obtained in the whole range of carrageenan concentrations investigated. At lower carrageenan concentrations, the calcite aggregates were observed, while at higher concentrations rhombohedral crystals of uniform size distribution were found. Spherical imprints on the surfaces of calcite particles precipitated at lower gel concentrations were observed, thus...

NMR of Na+, glycine and HDO in isotropic and anisotropic carrageenan gels

The chemical-interaction environment of several gels made from carbohydrate polymers consisting of unmodified commercial-grade ι- and κ-carrageenan and varying in their NaCl and polysaccharide concentrations was studied by solution-state 1H, 2H, and 23Na NMR spectroscopy. All gels showed isotropic NMR spectra at low gel concentrations (1% ‘w/w’) inside 10 mm NMR tubes. Anisotropy was introduced by stretching the gels, and the degree of alignment depended on the extent of stretching as well as gel and NaCl concentration. For Na+ a strong binding component of the anisotropy in ι- and less in κ-carrageenan gels was found, in contrast to a partial binding of glycine, and a spatial and gel-concentration dependent anisotropic effect for deuterated water (HDO). This finding is explained by the possibility of electrostatic interaction between Na+ and ionic sulfate groups in the carrageenan polymer, HDO probably only interacts via hydrogen bonding, while glycine could interact by both means. The lifetime of bound Na+ is likely to be small but bigger than for hydrogen-bonded HDO; thus Na+ would be more exposed to the electric field gradient tensor (caused by stretching the gel) than HDO, resulting in larger anisotropic spectral splitting and explaining the strong NaCl-concentration dependence of 23Na+ NMR anisotropy, which was not evident with HDO in 2H NMR spectra.

Experimental pattern transitions in a Liesegang system

We experimentally found various patterns of precipitation in a lead iodide Liesegang system: periodic precipitation, continuous tree-like crystal aggregates and periodic tree-like crystal aggregates. Gel and reactant concentrations are two major controlling factors. At high gel concentrations, Liesegang band forms, whereas at low gel concentrations, tree-like crystal aggregates similar to diffusion-limited aggregation (DLA) develop in the entire system. At an intermediate gel concentration, Liesegang band and tree-like crystal aggregates coexist. At lower reactant concentration, the periodic tree-like crystal aggregates tends to form. A simple scaling argument that the kinetics of bulk and surface nucleation plays an important role, is presented to understand the effect of gel concentration on pattern transition.

The rate of metal catalyzed oxidation of sulfur dioxide in collagen surrogates

Oxidation of sulfur dioxide (SO 2) in gelatine gels has been used as a model for the process in leather and other collagen artifacts. Oxidation rate constants for 5–30% w/w gelatine gels at 25 °C had a mean first order rate constant of 3.25 × 10 –6 s –1 (i.e. t 1 2 = 60 h). The water content of the gelatine has no effect on the observed rate of SO 2 oxidation. Copper can act as a catalyst, but is often rendered ineffective because it is bound to the gel. Significant rate increases only emerge at low gel concentrations and high free copper concentrations. Three copper ions bind to a single gel unit and it is all bound when the protein material (% w/w)/free copper (mol kg –1) value exceeds 100,000. The availability of free copper to promote catalysis is dramatically reduced as the gelatine concentration approaches 30% w/w. Concentrations of likely catalysts together with total sulfur were determined in 11 samples of historical parchment and leather using inductively coupled plasma (ICP) emission analysis: iron 0.058–1.513; copper 0.002–0.065; manganese 0.002–0.091; sulfur 1.962–28.273 (g kg –1 dry leather). Although sulfuric acid (5% w/w) causes degradation of gelatine, the presence of metals did not alter the rate of this process. Gelatine appeared to be a useful good surrogate for leather in this study.

DNA electrophoresis in agarose gels: effects of field and gel concentration on the exponential dependence of reciprocal mobility on DNA length.

Electrophoretic mobilities of DNA molecules ranging in length from 200 to 48 502 base pairs (bp) were measured in agarose gels with concentrations T = 0.5% to 1.3% at electric fields from E = 0.71 to 5.0 V/cm. This broad data set determines a range of conditions over which the new interpolation equation nu(L) = (beta+alpha(1+exp(-L/gamma))(-1) can be used to relate mobility to length with high accuracy. Mobility data were fit with chi(2) > 0.999 for all gel concentrations and fields ranging from 2.5 to 5 V/cm, and for lower fields at low gel concentrations. Analyses using so-called reptation plots (Rousseau, J., Drouin, G., Slater, G. W., Phys. Rev. Lett. 1997, 79, 1945-1948) indicate that this simple exponential relation is obeyed well when there is a smooth transition from the Ogston sieving regime to the reptation regime with increasing DNA length. Deviations from this equation occur when DNA migration is hindered, apparently by entropic-trapping, which is favored at low fields and high gel concentrations in the ranges examined.