Surface Dilational Rheology Research Articles

Nonlinear Surface Dilatational Rheology and Foaming Behavior of Protein and Protein Fibrillar Aggregates in the Presence of Natural Surfactant.

The surface and foaming properties of native soy glycinin (11S) and its heat-induced fibrillar aggregates, in the presence of natural surfactant steviol glycoside (STE), were investigated and compared at pH 7.0 to determine the impact of protein structure modification on protein-surfactant interfacial interactions. The adsorption at, and nonlinear dilatational rheological behavior of, the air-water interface were studied by combining drop shape analysis tensiometry, ellipsometry, and large-amplitude oscillatory dilatational rheology. Lissajous plots of surface pressure versus deformation were used to analyze the surface rheological response in terms of interfacial microstructure. The heat treatment generates a mixture of long fibrils and unconverted peptides. The presence of small peptides in 11S fibril samples resulted in a faster adsorption kinetics than that of native 11S. The addition of STE affected the adsorption of 11S significantly, whereas no apparent effect on the adsorption of the 11S fibril-peptide system was observed. The rheological response of interfaces stabilized by 11S-STE mixtures also differed significantly from the response for 11S fibril-peptide-STE mixtures. For 11S, the STE reduces the degree of strain hardening in extension and increases strain hardening in compression, suggesting the interfacial structure may change from a surface gel to a mixed phase of protein patches and STE domains. The foams generated from the mixtures displayed comparable foam stability to that of pure 11S. For 11S fibril-peptide mixtures STE only significantly affects the response in extension, where the degree of strain softening is decreased compared to the pure fibril-peptide system. The foam stability of the fibril-peptide system was significantly reduced by STE. These findings indicate that fibrillization of globular proteins could be a potential strategy to modify the complex surface and foaming behaviors of protein-surfactant mixtures.

Specific effects of Ca(2+) ions and molecular structure of β-lactoglobulin interfacial layers that drive macroscopic foam stability.

β-Lactoglobulin (BLG) adsorption layers at air-water interfaces were studied in situ with vibrational sum-frequency generation (SFG), tensiometry, surface dilatational rheology and ellipsometry as a function of bulk Ca(2+) concentration. The relation between the interfacial molecular structure of adsorbed BLG and the interactions with the supporting electrolyte is additionally addressed on higher length scales along the foam hierarchy - from the ubiquitous air-water interface through thin foam films to macroscopic foam. For concentrations <1 mM, a strong decrease in SFG intensity from O-H stretching bands and a slight increase in layer thickness and surface pressure are observed. A further increase in Ca(2+) concentrations above 1 mM causes an apparent change in the polarity of aromatic C-H stretching vibrations from interfacial BLG which we associate to a charge reversal at the interface. Foam film measurements show formation of common black films at Ca(2+) concentrations above 1 mM due to considerable decrease of the stabilizing electrostatic disjoining pressure. These observations also correlate with a minimum in macroscopic foam stability. For concentrations >30 mM Ca(2+), micrographs of foam films show clear signatures of aggregates which tend to increase the stability of foam films. Here, the interfacial layers have a higher surface dilatational elasticity. In fact, macroscopic foams formed from BLG dilutions with high Ca(2+) concentrations where aggregates and interfacial layers with higher elasticity are found, showed the highest stability with much smaller bubble sizes.

Complexation of phospholipids and cholesterol by triterpenic saponins in bulk and in monolayers

The interactions between three triterpene saponins: α-hederin, hederacoside C and ammonium glycyrrhizate with model lipids: cholesterol and dipalmitoylphosphatidylcholine (DPPC) are described. The oleanolic acid-type saponins (α-hederin and hederacoside C) were shown to form 1:1 complexes with lipids in bulk, characterized by stability constants in the range (4.0±0.2)·103–(5.0±0.4)·104M−1. The complexes with cholesterol are generally stronger than those with DPPC. On the contrary, ammonium glycyrrhizate does not form complexes with any of the lipids in solution. The saponin–lipid interactions were also studied in a confined environment of Langmuir monolayers of DPPC and DPPC/cholesterol with the saponins present in the subphase. A combined monolayer relaxation, surface dilational rheology, fluorescence microscopy and neutron reflectivity (NR) study showed that all three saponins are able to penetrate pure DPPC and mixed DPPC/cholesterol monolayers. Overall, the effect of the saponins on the model lipid monolayers does not fully correlate with the lipid–saponin complex formation in the homogeneous solution. The best correlation was found for α-hederin, for which even the preference for cholesterol over DPPC observed in bulk is well reflected in the monolayer studies and the literature data on its membranolytic activity. Similarly, the lack of interaction of ammonium glycyrrhizate with both lipids is evident equally in bulk and monolayer experiments, as well as in its weak membranolytic activity. The combined bulk and monolayer results are discussed in view of the role of confinement in modulating the saponin–lipid interactions and possible mechanism of membranolytic activity of saponins.

Comparison of the effect of two Quillaja bark saponin extracts on DPPC and DPPC/cholesterol Langmuir monolayers

The present study aims at comparing the effect of two commercially available Quillaja bark saponin (QBS) products on model Langmuir monolayers. Pure dipalmitoylphosphatidylcholine (DPPC) and mixed DPPC/cholesterol Langmuir monolayers were used as crude models of erythrocyte membranes in order to better understand a hemolytic activity of QBS. Both QBS products are mixtures of several saponins and non-saponin fractions, only few of which can be assigned an exact chemical structure, as shown by an HPLC analysis. Noticeable differences in the saponin profiles, and most importantly—in the content of non-saponin fractions (tannins and phenolic compounds) are probably responsible for their different adsorption behavior at the water/air interface. The lipids Langmuir monolayers were initially spread on pure water and compressed to surface pressure of 32.5mN/m, which is believed to provide the lipid packing similar to that in real biological membranes. The water subphase was subsequently exchanged with the respective QBS solutions in the concentration range 5×10−4 to 2×10−1wt%. In order to assess the resistance of the model lipid monolayers to QBS, a combination of surface pressure relaxation and surface dilatational rheology was employed. Both QBS are shown to penetrate the lipid layers without removing them, but their effect on the lipid layers' relaxation and viscoelastic dilational properties is different. The differences virtually disappear when cholesterol is present in the monolayer, especially for the DPPC/Cholesterol molar ratio between 10:9 and 4:1 (mol/mol). Despite several jumps of surface pressure, the lipid layers containing cholesterol can resist penetration of QBS.

Surface pressure affects B-hordein network formation at the air–water interface in relation to gastric digestibility

Protein interfacial network formation under mechanical pressure and its influence on degradation was investigated at molecular level using Langmuir–Blodgett B-hordein monolayer as a 2D model. Surface properties, such as surface pressure, dilatational and shear rheology and the surface pressure−area (π−A) isotherm, of B-hordein at air–water interface were analyzed by tensiometer, rheometer and a Langmuir–Blodgett trough respectively. B-Hordein conformation and orientation under different surface pressures were determined by polarization modulation-infrared reflection absorption spectroscopy (PM-IRRAS). The interfacial network morphology was observed by atomic force microscopy (AFM). B-Hordein could reduce the air–water surface tension rapidly to ∼45mN/m and form a solid-like network with high rheological elasticity and compressibility at interface, which could be a result of interactions developed by intermolecular β-sheets. The results also revealed that B-hordein interfacial network switched from an expanded liquid phase to a solid-like film with increasing compression pressure. The orientation of B-hordein was parallel to the surface when in expended liquid phase, whereas upon compression, the hydrophobic repetitive region tilted away from water phase. When compressed to 30mN/m, a strong elastic network was formed at the interface, and it was resistant to a harsh gastric-like environment of low pH and pepsin. This work generated fundamental knowledge, which suggested the potential to design B-hordein stabilized emulsions and encapsulations with controllable digestibility for small intestine targeted delivery of bioactive compounds.

Effect of Glyphosate Isopropylamine on the Surface Tension and Surface Dilational Rheology Properties of Polyoxyethylene Tallow Amine Surfactant

In order to develop the substitutes for polyoxyethylene tallow amine (POEA), the understanding the interaction of glyphosate isopropylamine and POEA is essential. The surface behaviors of POEA and POEA in the presence of 1 wt% glyphosate isopropylamine have been investigated at the air-water interface by the drop shape analysis method. The influences of surface tension, dilational frequency, and bulk concentration on the surface properties were expounded. The experiment results show that the adsorption films of POEA behave elastic in nature at low bulk concentration. With increasing in bulk concentration, the dilational modulus, dilational elasticity, and dilational viscosity pass through a maximum value, the phase angle increase monotonically. These phenomena can be attributed to the diffusion-exchange process between the bulk and the interface. The addition of 1 wt% glyphosate isopropylamine significantly influences on the POEA surface tension and dilational properties. The dilational modulus, dilational elasticity, and dilational viscosity obvious decrease in general, and the values of phase angle significant change after the addition of 1 wt% glyphosate isopropylamine. Glyphosate isopropylamine and POEA form a new complex in the solution and the surface activity and surface dilational properties of complex is different from POEA.

Adsorption of Proteins at the Solution/Air Interface Influenced by Added Nonionic Surfactants at Very Low Concentrations for Both Components. 3. Dilational Surface Rheology

The influence of the addition of the nonionic surfactants C12DMPO, C14DMPO, C10OH, and C10EO5 at concentrations between 10(-5) and 10(-1) mmol/L to solutions of β-casein (BCS) and β-lactoglobulin (BLG) at a fixed concentration of 10(-5) mmol/L on the dilational surface rheology is studied. A maximum in the viscoelasticity modulus |E| occurs at very low surfactant concentrations (10(-4) to 10(-3) mmol/L) for mixtures of BCS with C12DMPO and C14DMPO and for mixtures of BLG with C10EO5, while for mixture of BCS with C10EO5 the value of |E| only slightly increased. The |E| values calculated with a recently developed model, which assumes changes in the interfacial molar area of the protein molecules due to the interaction with the surfactants, are in satisfactory agreement with experimental data. A linear dependence exists between the ratio of the maximum modulus for the mixture to the modulus of the single protein solution and the coefficient reflecting the influence of the surfactants on the adsorption activity of the protein.

Surfactant induced complex formation and their effects on the interfacial properties of seawater

The effect of a cationic surfactant, hexadecyltrimethylammonium bromide (CTAB), on the interfacial properties of seawater has been studied by dynamic and equilibrium surface tension and by dilational rheology essays. Important modifications of the surface tension and dilational rheology response have been observed already at the very low CTAB concentrations, where the effects due to the high ionic strength are negligible. The comparison with the effects of CTAB in different seawater models, or in natural seawater fractions, points out the establishment of strong interactions between the surfactant molecules and the lipophilic fraction of organic material dispersed/dissolved in seawater, affecting the interfacial activity of the molecules.Considering the biochemical richness of seawater, these results can be explained assuming interaction mechanisms and adsorption schemes similar to those speculated for protein and other macromolecules in the presence of surfactants, which in fact show similar features. Thus already at the low concentrations the surfactant molecules form highly surface-active complexes with part of the organic fraction of seawater. At the larger surfactant concentrations these complexes compete for adsorption with an excess of free CTAB molecules which, according to the thermodynamic conditions, are most favoured to occupy the liquid interface.The results of this study underline the important role of the sea organic content in enhancing the surface-activity of surfactants, which is relevant for a deeper understand of the direct and indirect effects of these types of pollutants on the physico-chemical environment in the sea coastal areas and develop mitigation strategies.

Dynamic surface elasticity of the mixed solutions of bovine serum albumin and synthetic polyelectrolytes

The kinetics of dynamic surface elasticity of the mixed solutions of bovine serum albumin and synthetic polyelectrolytes makes it possible to apply dilational surface rheology to the determination of the adsorption mechanism of protein/polyelectrolyte complexes and the structure of adsorption layers.

Interfacial dynamic and dilational rheology of polyelectrolyte/surfactant two-component nanoparticle systems at air–water interface

The interfacial characteristics of nanoparticles and consequent inter-particle interactions at the interface are poorly understood. In this work, the interfacial dynamic and corresponding dilational surface rheology of self-assembled polyelectrolyte/surfactant nanoparticles at the air–water interface are characterized. The nanoparticles are prepared from dodecyltrimethylammonium (DTAB) and poly (sodium 4-styrene-sulfonate) (PSS) by mixing them in aqueous solution. The interfacial dynamic characteristics have been carried out by comparing the surface pressure with the dilational rheological response of these nanoparticles at interface. The results indicate that this type of nanoparticles can adsorb at the interface forming a nanoparticle monolayer, which leads to the surface tension decreased markedly. The dependence of surface pressure on time shows the instability and disassembly process of nanoparticles at the interface. On the basis of these observations, it is proposed that the nanoparticles undergo a dynamic process that interface induced nanoparticles disassembly into DTAB/PSS complexes. The presence of PSS in the subphase can promote the process of nanoparticles disassembly. A transition point in dilational elasticity and viscosity response of the nanoparticles versus oscillation frequency further validate the micro dynamic process of nanoparticles and the formation of polyelectrolyte/surfactant complex monolayer at the interface.

Synergistic foaming and surface properties of a weakly interacting mixture of soy glycinin and biosurfactant stevioside.

The adsorption of the mixtures of soy glycinin (11S) with a biosurfactant stevioside (STE) at the air-water interface was studied to understand its relation with foaming properties. A combination of several techniques such as dynamic surface tension, dilatational rheology, fluorescence spectroscopy, and isothermal titration calorimetry (ITC) was used. In the presence of intermediate STE concentrations (0.25-0.5%), the weak binding of STE with 11S in bulk occurred by hydrophobic interactions, which could induce conformational changes of 11S, as evidenced by fluorescence and ITC. Accordingly, the strong synergy in reducing surface tension and the plateau in surface elasticity for mixed 11S-STE layers formed from the weakly interacting mixtures were clearly observed. This effect could be explained by the complexation with STE, which might facilitate the partial dissociation and further unfolding of 11S upon adsorption, thus enhancing the protein-protein and protein-STE interfacial interactions. These surface properties were positively reflected in foams produced by the weakly interacting system, which exhibited good foaming capacity and considerable stability probably due to better response to external stresses. However, at high STE concentrations (1-2%), as a consequence of the interface dominated by STE due to the preferential adsorption of STE molecules, the surface elasticity of layers dramatically decreased, and the resultant foams became less stable.

Effect of temperature on surface tension and surface dilational rheology of type I collagen

The dynamic surface tension and surface dilational rheology of native collagen (type I, from bovine calf skin) were analyzed in the temperature range from 21 to 34.5°C. In addition, the effect of heating the collagen solutions up to 90°C followed by cooling down to room temperature, on surface tension and rheological parameters of the adsorbed layers was studied. As a water-soluble fibrous protein with little exposed hydrophobicity, native collagen is weakly surface active. However, its layers formed by spontaneous adsorption on the water/air surface display exceptional dilational rheology parameters, comparable with those of globular proteins: at high frequency limit (0.1Hz) the layers are predominantly elastic, with the storage modulus, E′=58mN/m for collagen concentration of 1.3×10−5M (4g/L). Increasing temperature during adsorption helps to reduce the surface tension, but the resulting layers show diminished surface elastic response. On the other hand, the samples heated at elevated temperatures and measured after cooling down to the room temperature maintain their high surface elasticity. The results suggest that collagen, even in its native form (not processed to gelatin) has a potential to stabilize biocompatible foams/emulsions by forming mechanically strong adsorbed layers.

Interfacial layer properties and foam film drainage kinetics of aqueous solutions of hexadecyltrimethylammonium chloride

Comparative studies of adsorption layer properties at the air/solution interface – dynamic and equilibrium surface tension measurements, surface dilatational rheology – and the drainage behaviour of microscopic foam films of aqueous solutions of hexadecyltrimethylammonium chloride (cetyltrimethylammonium chloride, CTAC), are performed within a broad surfactant concentration range. Two different concentrations of the added electrolyte (sodium chloride, NaCl) are investigated: 0.1M and 0.5M. The interfacial layer characteristics are studied using profile analysis tensiometer (PAT-1). The foam film drainage kinetics is investigated by the microinterferometric technique of Scheludko–Exerowa. Concerted onset of specific peculiarities is established for both adsorption layer properties and the foam film parameters. A remarkably good correlation is established between the run of the dynamic and equilibrium surface tension isotherms, the course of the surface dilatational elasticity and the peculiarities of drainage kinetics and film stability against the surfactant concentration within each of these concentration intervals. The obtained results shed additional light on the important role of the surfactant adsorption layers for understanding the details of microscopic foam film drainage and stability characteristics.

Unusual penetration of phospholipid mono- and bilayers by Quillaja bark saponin biosurfactant

The interactions between a model phospholipid 1,2-dipalmitoyl-sn-glycero-3-phosphocholine (DPPC) and a biosurfactant Quillaja Bark Saponin (QBS) obtained from the bark of Quillaja saponaria Molina were studied using simple models of biological membranes. QBS is known to interact strongly with the latter, exerting a number of haemolytic, cytotoxic and anti-microbial actions. The interaction of QBS dissolved in the subphase with DPPC monolayers and silicon-supported bilayers was studied above the cmc (10−3M). Surface pressure relaxation and surface dilatational rheology combined with quartz crystal microbalance (QCM) and neutron reflectivity (NR) were employed for this purpose. The DPPC-penetrating abilities of QBS are compared with those of typical synthetic surfactants (SDS, CTAB and Triton X-100). We show that the penetration studies using high surface activity (bio)surfactants should be performed by a subphase exchange, not by spreading onto the surfactant solution. In contrast to the synthetic surfactants of similar surface activity, QBS does not collapse DPPC mono- and bilayers, but penetrates them, improving their surface dilatational elastic properties even in the highly compressed solid state. The dilatational viscoelasticity modulus increases from 204 mN/m for pure DPPC up to 310 mN/m for the QBS-penetrated layers, while it drops to near zero values in the case of the synthetic surfactants. The estimated maximum insertion pressure of QBS into DPPC monolayers exceeds the maximum surface pressure achievable in our setup, in agreement with the surface rheological response of the penetrated layers.

Effect of hydration of sugar groups on adsorption of Quillaja bark saponin at air/water and Si/water interfaces

Adsorption of a natural glycoside surfactant Quillaja bark saponin (“QBS”, Sigma Aldrich 84510) was studied at the air/water and Si/water interfaces using a combination of surface pressure (SP), surface dilatational rheology, neutron reflectivity (NR), Infra-Red Attenuated Total Reflection Spectroscopy (IR ATR) and Quartz Crystal Microbalance (QCM). The adsorbed layers formed at the air/water interface are predominantly elastic, with the dilatational surface storage modulus reaching the maximum value of E′=184mN/m. The NR results point to a strong hydration of the adsorbed layers (about 65% hydration, corresponding to about 60 molecules of water per one QBS molecule), most likely related to the presence of multiple sugar groups constituting the glycone part of the QBS molecules. With a layer thickness of 19Å, the adsorbed amount obtained from NR seems largely underestimated in comparison to the value obtained from the surface tension isotherm. While this high extent of hydration does not prevent formation of dense and highly elastic layers at the air–water surface, QBS adsorption at the Si/water interface is much weaker. The adsorption isotherm of QBS on Si obtained from the QCM study reflects much lower affinity of highly hydrated and negatively charged saponin molecules to the Si/water interface. We postulate that at the air/water interface, QBS adsorbs through the triterpene aglycone moiety. In contrast, weak hydrogen bonding between the glycone part and the surface silanol groups of Si is responsible for QBS adsorption on more polar Si/water interface.

Surface and foaming properties of polyoxyethylene glycerol ester surfactants

Polyoxyethylene glycerol esters derived from cocoa oil are non-ionic surfactants obtained from a renewable source which fulfil the environmental and toxicological requirements to be used as ecofriendly foaming and/or emulsifying agents. This paper reports a study on the equilibrium adsorption, surface rheology and foaming properties of two commercial polyoxytheylene glycerol ester surfactants which differ in the number of ethylene oxide (EO) groups. Dynamic and equilibrium surface pressure values were obtained with a drop profile tensiometer. The oscillating drop technique was used to study the dilatational rheology of adsorbed surfactant layers. The foaming properties of aqueous solutions of these surfactants were characterized in a commercial foam scan column. Dynamic surface pressure measurements showed two adsorption processes in which a slower condensation of the adsorbed film followed a fast diffussion step. Surface rheology and equilibrium surface pressure data fitted an extended Frumkim model which takes into account the compressibility of the adsorbed layer. The polyoxyethylene glycerol ester with the highest number of EO groups turned out to be more surface active, leading to lower cmc and higher adsorption constant. Both surfactants exhibited similar overall foam capacity at a given concentration. However, the time evolution of the liquid volume in the foam and the size of the air bubbles pointed to the formation of a more homogeneous and stable foam with smaller bubbles for the most hydrophilic polyoxyethylene glycerol ester studied. This is considered to be related to its surface dilatational rheology properties.

Ionic Liquid Doped β Lactoglobulin as Template for Nanoclusters of Nickel Oxide

In this work, Langmuir films of organized assemblies of β-lactoglobulin (βLG) with 1-ethyl-3-methyl imidazolium ethyl sulfate (IL-emes) have been characterized at air/water interface using surface pressure-specific area isotherms and dilational rheology. The protein in the IL-mediated assembly shows excellent packing at the interface and is stable as seen in circular dichroic spectroscopy. These spread films on nickel chloride were transferred as Langmuir–Schaffer films of βLG and βLG+IL-emes and used as template for designing nanoclusters of nickel oxide. The nanoclusters have been characterized using transmission electron microscopy (TEM) and powder XRD. While pure protein template gives needle-shaped structures, the IL-mediated template gives spherical shapes of hexagonal nickel oxide in the range 30 nm to 40 nm. Presence of ionic liquid seems to slow down the growth of the cluster and also prevents aggregation of the clusters.

Hydrophobic modification of dextran with 1,2-epoxyalkanes in aqueous micellar medium: Competition between interfacial and bulk reactions and consequences on polymer properties

Hydrophobic modification of dextran (degree of polymerization around 125) by reaction with 1,2-epoxydodecane was investigated in micellar aqueous medium in which epoxide was dispersed under magnetic stirring in the presence of a cationic surfactant. A base catalyst, sodium hydroxide, was added in water. Dextran derivatives with degrees of substitution up to 10% were prepared. Apart from dextran modification, epoxide molecules underwent interfacial oligomerization which led to by-products that had to be extracted by dialysis followed by Soxhlet extraction. Two main reaction parameters influenced the amount of oligomeric by-products, i.e. temperature and surfactant structure. Adsorption of these compounds at air/water interface was studied by surface tension and dilational rheology measurements and compared to that observed with amphiphilic dextrans prepared in homogeneous conditions.

Interfacial characterization of Pluronic PE9400 at biocompatible (air–water and limonene–water) interfaces

In this work, we provide an accurate characterization of non-ionic triblock copolymer Pluronic PE9400 at the air–water and limonene–water interfaces, comprising a systematic analysis of surface tension isotherms, dynamic curves, dilatational rheology and desorption profiles. The surface pressure isotherms display two different slopes of the Π–c plot suggesting the existence of two adsorption regimes for PE9400 at both interfaces. Application of a theoretical model, which assumes the coexistence of different adsorbed states characterized by their molar areas, allows quantification of the conformational changes occurring at the adsorbed layer, indentifying differences between the conformations adopted at the air–water and the limonene–water interface. The presence of two maxima in the dilatational modulus vs. interfacial pressure importantly corroborates this conformational change from a 2D flat conformation to 3D brush one. Moreover, the dilatational response provides mechanical diferences between the interfacial layers formed at the two interfaces analyzed. Dynamic surface pressure data were transformed into a dimensionless form and fitted to another model which considers the influence of the reorganization process on the adsorption dynamics. Finally, the desorption profiles reveal that Pluronic PE9400 is irreversibly adsorbed at both interfaces regardless of the interfacial conformation and nature of the interface. The systematic characterization presented in this work provides important new findings on the interfacial properties of pluronics which can be applied in the rational development of new products, such as biocompatible limonene-based emulsions and/or microemulsions.

Effect of tea polyphenols on the dilational rheology of Human Whole Saliva (HWS): Part 1, HWS characterization

Human whole saliva (HWS) is a physiological secretion that performs a number of essential functions including protection of oral health, lubrication of mouth tissues and food pre-digestion. The complexity of the saliva proteome reflects the unique physiochemical functionality of this biofluid. In addition, saliva influences the transduction of aroma, taste and tactile (mouthfeel) perceptions through the interaction with food and beverages. This work, which is presented as two distinct contributions, reports on the extensive interfacial properties characterization (air–liquid) of saliva either alone or after exposure to tea polyphenols. This first part discusses the complex nature of the salivary film in terms of kinetic and viscoelastic (dilational surface rheology) properties. The formation of the surface film is analyzed with time and explained in terms of the progressively irreversible adsorption of protein fractions of larger molecular weight. The viscoelastic response to appropriate frequency modulation is modelled using a general mixed kinetic model, which satisfactorily reproduces the experimental data in better agreement than the traditional Lucassen diffusive model.