Rigid Sphere Model Research Articles

Volumetric, ultrasonic, viscometric and refractive index behavior of binary mixtures of 2,2,4-trimethylpentane with aromatic hydrocarbons: An experimental and theoretical study

Measurements of densities, ρ, ultrasonic speeds, u, viscosities, η, and refractive indices, n have been carried out for binary mixtures of 2,2,4-trimethylpentane (TMP) with benzene, toluene, o-xylene, m-xylene, and p-xylene, including those of pure liquids, at 35 °C over the entire composition range. From these experimental data, the values of excess molar volumes, V E, deviations in isentropic compressibilities, Δ k s, and viscosities, Δ η, excess Gibbs free energy of activation of viscous flow, G ⁎ E, deviations in refractive indices, Δ n, and partial molar isentropic compressibility, K¯ 2°, of aromatic hydrocarbons in TMP at infinite dilution were calculated. The variations of these parameters with composition of the mixtures suggest an increasingly better packing of plate-like aromatic molecules into the spaces created by globular-shaped TMP molecules in the sequence: benzene < toluene < m-xylene ≈ p-xylene < o-xylene. Further, for xylenes, the magnitude of these derived parameters is found to depend on the position of –CH 3 groups in the molecule. The excess molar volumes, V E, of the mixtures have also been predicted from Flory's statistical theory and the Prigogine–Flory–Patterson (PFP) theory. Using Flory theory and rigid hard sphere models, the values of isothermal compressibilities, k T, of the present binary mixtures were evaluated and compared with the experimental values. Moreover, theoretical evaluation of viscosity by using one-, two- and three-parameter equations suggests the superiority of the three-parameter equation over one- and two-parameter equations.

First-Principles Calculations of Migration Energy of Lithium Ions in Halides and Chalcogenides

Migration of Li+ ions via the vacancy mechanism in LiX (X = F, Cl, Br, and I) with the rocksalt and hypothetical zinc blende structures and Li2X (X = O, S, Se, and Te) with the antifluorite structure has been investigated using first-principles projector augmented wave calculations with the generalized gradient approximation. The migration paths and energies, determined by the nudged-elastic-band method, are discussed on the basis of two idealized models: the rigid-sphere and charged-sphere models. The trajectories and energy profiles of the migration in these lithium compounds vary between these two models, depending on the anion species and crystal structure. The migration energies in LiX with both the rocksalt and hypothetical zinc blende structures show a tendency to decrease with increasing periodic number of the anion species in the periodic table. This is consistent with the widely accepted view that anion species with large ionic radii and high polarizabilities are favorable for good ionic conduction. In contrast, Li2O exhibits the lowest migration energy among Li2X compounds, although O is the smallest among the chalcogens, indicating that electrostatic attractive interactions play the dominant role in the inter-ion interactions in Li2O and, therefore, in the ion migration.

Dynamics of Vesicles in a Wall-Bounded Shear Flow

We report a detailed study of the behavior (shapes, experienced forces, velocities) of giant lipid vesicles subjected to a shear flow close to a wall. Vesicle buoyancy, size, and reduced volume were separately varied. We show that vesicles are deformed by the flow and exhibit a tank-treading motion with steady orientation. Their shapes are characterized by two nondimensional parameters: the reduced volume and the ratio of the shear stress with the hydrostatic pressure. We confirm the existence of a force, able to lift away nonspherical buoyant vesicles from the substrate. We give the functional variation and the value of this lift force (up to 150 pN in our experimental conditions) as a function of the relevant physical parameters: vesicle-substrate distance, wall shear rate, viscosity of the solution, vesicle size, and reduced volume. Circulating deformable cells disclosing a nonspherical shape also experience this force of viscous origin, which contributes to take them away from the endothelium and should be taken into account in studies on cell adhesion in flow chambers, where cells membrane and the adhesive substrate are in relative motion. Finally, the kinematics of vesicles along the flow direction can be described in a first approximation with a model of rigid spheres.

Temperature Dependence of Viscosities of Common Carrier Gases

The temperature dependence of gas viscosities deviates significantly from the rigid-sphere model prediction of T1/2. A better relationship of Tx, where x > 1/2, is described on the basis of two the...

Electrostatic interaction models for ion binding to humic substances

Preferably, the description of ion binding to humic substances (HSs) is done with thermodynamic constants that do not depend on the environmental conditions. To solve this problem, models have to be made that describe the electrostatic and specific interactions. With a given electrostatic model the charge/pH curves of HS at different salt levels can be re-plotted as a function of the local pH near the sites (pH loc) of HSs. If the model is appropriate, the charge/pH loc curves will merge into a master curve (MC). In this study five electrostatic models were investigated to obtain pH loc for purified Aldrich humic acid (PAHA): rigid sphere (RS), ion-permeable sphere, Donnan (NICA), Donnan-EV, and Donnan-EDL. The RS model is tested in two versions; one based on the measured hydrodynamic radius (RS- a h) and the other on an optimized radius (RS- a opt). The ion-permeable sphere model uses the hydrodynamic radius and provides the potential distribution; as characteristic potential the radial average inside the sphere is used. In the Donnan (NICA) model the volume of the sphere, V D, in which the charge of PAHA is neutralized, is optimized with a constraint between V D and ionic strength, and in the Donnan-EV model V D is calculated by setting the radius of the gel as the sum of the hydrodynamic radius of PAHA and the Debye length. The Donnan-EDL model uses the hydrodynamic particle radius and is based on a combination of the Donnan model and the diffuse electrical double layer model. Only the RS- a opt, Donnan (NICA), and Donnan-EV models give adequate MCs. The positions of the MCs differ with respect to each other. This means that the discrimination between electrostatic and intrinsic interactions is model-dependent and therefore arbitrary. The Donnan (NICA) model has a practical advantage over the other two models because this model needs no measurements of the size of HS. For the purpose of the routine fitting of ion-binding data to an ion-binding isotherm equation that includes the electrostatics, this advantage is quite important.

Evidence for Rigid Binding of Rhodamine 6G to Silica Surfaces in Aqueous Solution Based on Fluorescence Anisotropy Decay Analysis

Strong ionic binding of the cationic probe rhodamine 6G (R6G) to the anionic surface of silica particles in water provides a convenient labeling procedure to study both particle growth kinetics and surface modification by time-resolved fluorescence anisotropy (TRFA). The decays for R6G dispersed in diluted Ludox silica sols usually fit to a sum of picosecond and nanosecond decay components, along with a significant residual anisotropy component. The origin of the nanosecond decay component (phi2) is not fully understood, and has been ascribed to wobbling of the probe on the silica surface, the presence of a subpopulation of small nanoparticles in the Ludox sol, or rapid exchange between free and bound R6G. To elucidate the physical meaning of phi2, measurements were performed in various silica-based colloidal systems using different concentrations of silica. We found that the fraction of phi2 was generally higher in Ludox than in aqueous sodium silicate and decreased with increasing silica concentration; phi2 vanished upon gelation of sodium silicate at pH 7 leading to a total loss of R6G depolarization (r(t) = const). These results rule out the presence of local R6G wobbling when bound ionically to colloidal silica and support the rigid sphere model to describe the TRFA decays for R6G-Ludox. This conclusion is entirely supported by steady-state anisotropy data and structural considerations for the R6G molecule and the silica surface.

Nuclear Magnetic Relaxation of Oligo- and Poly(α-methylstyrene)s in Dilute Solution

The spin–lattice relaxation time T1 and nuclear Overhauser enhancement NOE were determined for three samples of atactic oligo- and poly(α-methylstyrene)s (a-PαMS) with weight-average degree of polymerization xw = 2, 3, and 67.1 in cyclohexane at 30 °C. A comparison is made of the present data for T1 and NOE with the helical wormlike (HW) chain theory, and it is shown that the theory may explain well the data for large xw (≳ 10). For smaller xw, the rigid sphere model having the radius equal to the apparent root-mean-square radius of gyration of the HW chain fails to give a quantitative explanation of T1 in contrast to the case of the relaxation time τΓ determined from dynamic depolarized light scattering measurements, indicating that the nuclear magnetic relaxation and dynamic depolarized light scattering cannot be described in terms of a common single relaxation time. However, it is shown that there is an effective (mean) magnetic relaxation time τM approximately equal to 0.6τΓ. From a comparison of the present results for T1 for a-PαMS with previous ones for typical flexible polymers, it is also shown that the behavior of T1 for them, including a-PαMS, may be consistently explained by the use of τM (proportional to τΓ).

Dynamic Depolarized Light Scattering by Oligo- and Poly(α-methylstyrene)s in Dilute Solution

The power spectrum JΓ of the excess depolarizedcomponent of scattered light was determined from frequency-domain dynamic depolarized light scattering measurements for 10 samples of atactic oligo- and poly(α-methylstyrene)s (a-PαMS), each with the fraction of racemic diads fr=0.72, in the range of weight-average degree of polymerization xw from 2 to 67.1 in cyclohexane at 30.5 °C (Θ). It is found that, as in the cases of atactic polystyrene (a-PS), atactic poly(methyl methacrylate) (a-PMMA), and isotactic (i-) PMMA, JΓ may be well represented in terms of a single Lorentzian independently of xw, and that the relaxation time τΓ as defined as the reciprocal of the half-width at half-maximum of JΓ evaluated at infinite dilution increases with increasing xw and levels off to its asymptotic value in the limit of xw→∞, being consistent with the theoretical prediction on the basis of the helical wormlike (HW) chain model. A comparison is made of the present data for τΓ with the HW theory, and it is shown that the theory may explain satisfactorily the data in the range of xw ≳ 10, although semiquantitatively. For xw ≲ 10, the rigid sphere model having the radius equal to the apparent root-mean-square radius of gyration of the HW chain may give a good explanation of τΓ. From a comparison of the present results for τΓ for a-PαMS with previous ones for a-PS and a- and i-PMMAs, it is shown that there is good correlation between static and dynamic chain stiffness, the latter being defined as the ratio of the relaxation time associated with the local motion of the long chain to that of its isolated repeat unit (monomer), as predicted by the HW theory.

Velocity slip and temperature jump coefficients for gaseous mixtures. IV. Temperature jump coefficient

The temperature jump coefficient was calculated for a binary gaseous mixture on the basis of the McCormack kinetic model of the Boltzmann equation, which was solved by the discrete velocity method. The calculations were carried out for the three mixtures of noble gases: neon–argon, helium–argon and helium–xenon. A strong influence of the intermolecular interaction on the temperature jump coefficient was observed by comparing the results based on the model of rigid spheres with those obtained for a realistic potential.

Formability and thermal stability of α′ phase in (Fe1−y Coy)–(B, C, N) films

AbstractIn order to find a way to obtain stable α″‐Fe16X2 phase, the formability and thermal stability of α′ (bct) phase were discussed. According to a rigid sphere model, we concluded that the less formability of B for the α′ phase is due to its large atomic radius. We elucidated the difference of thermal stability of α′‐Fe–X, taking into account their decomposition process. While, the decomposition of α′‐Fe–N progresses only by the migration of N, without changing the bone structure of Fe lattice, the additional energy is needed to break the original α‐Fe lattice in the cases of α′‐Fe–B and α′‐Fe–C. Therefore thermal stability of α′‐Fe–B and α′‐Fe–C is higher than that of α′‐Fe–N. (© 2004 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Velocity slip and temperature jump coefficients for gaseous mixtures. II. Thermal slip coefficient

The thermal slip coefficient was calculated for a binary gaseous mixture on the basis of the McCormack kinetic model of the Boltzmann equation, which was solved by the discrete velocity method. The calculations were carried out for the three mixtures of noble gases: neon–argon, helium–argon, and helium–xenon. A strong influence of the potential of intermolecular interaction upon the thermal slip coefficient was observed by comparing the results based on the model of rigid spheres with those obtained for a realistic potential. An example of application of the thermal slip coefficient is given.

Gaseous mixture flow between two parallel plates in the whole range of the gas rarefaction

The flow of binary gaseous mixtures between two parallel plates driven by gradients of pressure, temperature and concentration is studied, based on the McCormack model of the Boltzmann equation. The coupled kinetic equations are solved numerically by the discrete velocity method. The mass flow, the heat flux and the diffusion flux, which are the mixture quantities of practical importance, are expressed in terms of the so-called thermodynamic fluxes. The latter are written in a form that allows us to verify the Onsager–Casimir reciprocity relations. In addition, analytical expressions for these quantities are derived in the limit case of the hydrodynamic regime. Thus, the numerical solution together with these expressions provides the solution in the whole range of the gas rarefaction. The influence of the intermolecular interaction potential is also investigated by comparing the results for the rigid sphere model with those for a realistic potential. Numerical results are presented for two binary mixtures of noble gases (Ne–Ar and He–Xe) for various values of the molar concentrations.

A New Model for Calculating the Viscosity of Pure Liquids at High Pressures

In this paper we present a new model for correlating the dynamic viscosity of Newtonian liquids at high pressures. The proposed model is based on Eyring's absolute rate theory for liquid viscosity and considers the activation energy for viscous flow as being a thermodynamic free energy. The viscosity of the system is described by a combination of a reference term, given by the Chapman−Enskog theory, and a deviation contribution. A virial-type expansion in pressure and a term comprising an expression for the residual Helmholtz free energy of the system account for the deviation from the nonattracting rigid sphere model viscosity behavior. Three cubic equations of state, Peng−Robinson, Soave−Redlich−Kwong, and Peng−Robinson−Stryjek−Vera, have been tested for evaluating the residual Helmholtz free energy. The model requires only two adjustable parameters for each pure liquid, at each temperature. The parameters have been determined by fitting literature viscosity data of 49 different pure liquid compounds from pressures of 0.1−250 MPa within the reduced temperature interval of 0.4−0.7. The performance of the model has been found to be insensitive to the choice of the equation of state, except at pressures above 100 MPa for which only the Soave−Redlich−Kwong equation of state has been able to describe the volumetric behavior of the liquids. The studied liquids are n-alkanes, substituted alkanes, n-alkenes, cyclic alkanes, aromatics, alcohols, esters, 1-butylamine, argon, nitrogen, oxygen, ammonia, and water. The calculated values are in good agreement with the experimental ones. The value of the overall average absolute deviation, for the 4380 data points correlated with the present model, is 1.22%.

Structural phase transition of ScSb and YSb with a NaCl-type structure at high pressures

By use of synchrotron radiation, powder X-ray diffraction of ScSb and YSb with a NaCl-type structure has been studied up to 45 GPa at room temperature. A first-order phase transition from the NaCl-type (B1) to a CsCl-type structure (B2) began to occur at around 28 GPa for ScSb and at around 26 GPa for YSb. Crystal data of the high-pressure phase of both antimonides are obtained. The high-pressure structural behavior of ScSb and YSb is similar to that of heavier LnSb (Ln=Dy–Lu). The B1–B2 transition for ScSb and YSb can be understood according to the rigid sphere model. The bulk moduli of ScSb and YSb are about 58 GPa at ambient pressure.

Influence of Shots' Material on Shot Peening, a Finite Element Model

Shot peening is a mechanical surface treatment widely used in industry to improve the fatigue life of metal components. During the process, hard shots hit the surface of the treated mechanical part at high velocity, locally introducing a residual stress field due to plastification effects. To optimize the process, several analytical and numerical models have been proposed. Existing simulations show their efficiency to better understand the mechanism of residual stress introduction and better evaluate the influence of the shot peening parameters on the residual stress profile. Because the shots used in shot peening are always harder than the treated material itself, a classical approach consists in using a rigid sphere model in the computation; it is indeed usually necessary to introduce simplifying hypotheses due to the relative complexity of the process. During the industrial process, the shots may, in fact, be elastically and plastically deformed. Then, the impact will print a weaker dent in the material and the residual stress profile will be affected accordingly. Thus, it seems interesting to quantify these effects and we propose a finite element model for shot peening where the influence of the parameters associated to the shots, that is to say the velocity, radius and yield stress, are carefully analyzed. To better understand the process, we propose to compare the results of finite element models in which several sets of material parameters have been considered for the shots and the treated part. Several values of the shots' velocity and radius, chosen among the values commonly used in industry are considered. The resulting residual stress profiles are presented. It appears that the residual stress profiles in the solid are indeed sensitive to the shots' yield stress; this effect can be significant.

Ion beam-induced enhanced adhesion of gold films deposited on glass

The metallisation of glass for decorative and/or functional purposes is now a well-established technique. The most popular methods are electroplating or sputter deposition. In order to obtain suitable adhesion, substrate pretreatment is a substantial part of a vapour-deposited coating that generally cannot be dispensed with. The pretreatment costs can reach the same order of magnitude as those associated with the actual coating process, or can even exceed them. In this work we studied the adhesion of Au thin films on glass. The substrates were pretreated by an ion-beam-mixing step, consisting of the deposition of Au/C bilayers or C/Au/C multilayers followed by Xe + implantation. After such preparation, the specimens were further coated (using a sputtering machine) with 150-nm-thick Au or Au-alloy films. Adhesion properties of the films were examined using a scratch tester in conjunction with scanning electron microscopy. It was observed that, without the ion-beam-mixing pretreatment, the coatings were poorly adherent. Strong adhesion enhancement was observed in the pretreated samples. The key mechanism envisaged to explain this is related to the formation of mixed SiC–Au phases at the interface region. Moreover, the mechanical properties of the pure and alloyed Au films were quantified by nano-indentation, and hardness results are in good agreement with a simple rigid-sphere model of substitutional hardening.

A rigid sphere model for the melting of argon

A model is proposed for liquid and solid argon, in which the repulsive forces are represented by rigid cores and the attractive forces by a uniform background potential which depends on density but not on temperature. Thermodynamic consistency implies the equation of state: where p0 is the pressure exerted by rigid spheres at the same number density n and temperature T, and a is independent of n and T. When p0 is taken from the calculations of Alder and Wainwright it becomes possible to make an absolute calculation of the values of several dimensionless thermodynamic quantities characteristic of the triple point. All of these except certain second derivatives of the free energy agree very well with experiment, suggesting that the structure of liquids and solids near the triple point is mainly determined by the repulsive forces, the attractive forces serving merely to keep the molecules together.

The temperature-jump problem for a variable collision frequency model

An analytical version of the discrete-ordinates method is used here in the field of rarefied-gas dynamics to solve a version of the temperature-jump problem that is based on a linearized, variable collision frequency model of the Boltzmann equation. In addition to a complete development of the discrete-ordinates method for the application considered, the computational algorithm is implemented to yield accurate numerical results for three specific cases: the classical BGK model, the Williams model (the collision frequency is proportional to the magnitude of the velocity), and the rigid-sphere model.

Boundary slip phenomena in a binary gas mixture

The slip phenomena in gas mixtures are of fundamental significance in the specification of boundary conditions for flows in the slip regime. In a recent paper, new explicit results for the slip coefficients appropriate to binary gas mixtures were reported. The present work being reported extends the previous work to a higher level of accuracy by involving a higher order Chapman-Enskog expansion. In particular, new expressions for the slip coefficients are presented which are applicable for arbitrary models of the intermolecular interaction. Limiting expressions for the slip coefficients are given (for a simple gas) and the accuracy of the theory is discussed. Numerical calculations of the slip coefficients for different binary gas mixtures using the first and second order Chapman-Enskog approximations and the rigid sphere and Lennard-Jones (12-6) potential models have been carried out. The thermal creep and diffusion slip coefficients are found to be sensitive to the order of the approximation and to the potential model used. A comparison of the new higher order results with some of our previously obtained experimental data for the thermal transpiration effect has also been carried out and shows excellent agreement between the theory and the experiments which confirms the accuracy of the theory.

Extraction of rare earth metals with a multistage mixer-settler extraction column

Extraction behavior of rare earth metals within a mixer–settler extraction column has been analyzed with the stage efficiency calculated from mass transfer coefficients and interfacial area. The mass transfer coefficient within the dispersed drops is determined from a rigid sphere model by taking into account the residence time distribution of drops, and the coefficient around the drops is calculated by Ranz–Marshall's correlation with the terminal settling velocity of a rigid sphere. The interfacial area of dispersed drops is calculated by the use of correlations for the drop diameter and the holdup of dispersed phase in the mixer–settler extraction column. The calculated results for the separation of samarium and gadolinium with a five-stage mixer–settler extraction column are compared with the experimental results at various agitation speeds and flow ratios between two phases. The extraction behavior in the multistage column is explained by a model based on the hydrodynamics and the mass transfer within the mixer. Effects of the pH value in aqueous phase, the extractant concentration in organic phase and the number of stages on the extraction behavior in the mixer–settler column are also shown.