Equilibrium Concentration Profiles Research Articles

Coarsening of random interfaces in the spinodal decomposition of a binary fluid

The growth of random interfaces during the late stage spinodal decomposition for a near symmetric quench of a binary fluid is analyzed. Inertial effects are neglected, and the motion of the interface is determined by a balance between the surface tension, which tends to reduce the curvature, and the viscous stresses in the fluid. The interface is described by an “area distribution function” A(K,t), defined so that A(K,t)dKdx is the area of the interface with curvature in the interval dK about K in the volume dx at time t. Here, K=(K12+K22)1/2 is the magnitude of the curvature, and K1 and K2 are the principal curvatures. There is a change in the area distribution function due to a change in the curvature, and due to the tangential compression of the interface. Phenomenological relations for the change in curvature and surface area are obtained using the assumption that the only length scale affecting the dynamics of the interface at a point is the radius of curvature at that point. These relations are inserted in the conservation equation for the interface, and a similarity solution is obtained for the area distribution function. This solution indicates that the area of the interface decreases proportional to t−1 in the late stages of coarsening, and the mean curvature also decreases proportional to t−1. The effect of the motion of the interface on the interfacial concentration profile and interfacial energy is analyzed using a perturbation analysis. The diffusion equation for the concentration in the interfacial region contains an additional source term due to the convective transport of material caused by the motion of the interface, and this causes a correction to the equilibrium concentration profile of the interface. The excess interfacial energy due to the nonequilibrium motion of the interface is calculated using the Cahn–Hilliard square gradient free energy for a near-critical quench. It is found that the variation in the concentration causes an increase in the interfacial energy which is proportional to the curvature K of the interface.

Rheology of Colloidal Particles in a Confined Channel under Shear Flow by Brownian Dynamic Simulations

A colloidal system of strongly charged particles, confined between two charged walls, is studied under static conditions and in the presence of shear flows. Brownian Dynamics simulations (BD) are used to determine the concentration profiles for several separations of charged walls. Results show good agreement with those obtained under static conditions using the Hypernetted Chain Approximation (HNC) and Monte Carlo simulations. Results obtained with BD and HNC for neutral walls show more similarities than those between BD and Monte Carlo simulations depending on the initial state of the colloidal particles. The presence of a shear flow field perturbs the equilibrium concentration profiles and the distribution function in the flow direction, generating a structureless system, as confirmed by the absence of peaks in the radial distribution function. The mobility of the particles in the transversal direction decreases rapidly and becomes practically frozen. The flow exhibits a non-Newtonian behavior with shear-thinning viscosity. Due to the interparticle interactions and particle–wall interactions, the viscosity is lower as the wall separation decreases, giving rise to an apparent slip in the colloidal suspension. The slip velocity for repulsive walls is higher than that obtained with neutral walls and increases with the shear stress according to a power law, as observed in polymer solutions. The shear viscosity and the normal stress differences depend strongly on the combined effect of confinement, concentration of particles, magnitudes of inter-particle interactions and wall–particle repulsion.

First-principles approaches to surface segregation

Abstract The progress in the first-principles determination of the thermal equilibrium concentration profile at the surface of a random binary alloy is reviewed. After a brief introduction to the (single-site) coherent-potential approximation used to model concentrated alloys, and to the elements of density functional theory necessary for the ab initio treatment of such systems, three self-consistent Green function techniques for calculating ground-state properties of random alloy surfaces, namely the layer Korringa-Kohn-Rostoker (KKR), the screened KKR and the tight-binding linear muffin-tin orbital (LMTO) method are presented, with particular emphasis on the relationship between the latter two approaches. Although each of these methods is capable of yielding the internal energy of the system for any given profile, a ‘brute force’ search for the optimum configuration is unrealistic, in view of the large number of possibilities available. Instead, one uses the results of the above self-consistent calculations to construct an effective Ising model, which can then be treated by the standard methods of classical statistical mechanics. The two schemes currently used to generate the parameters of this effective Ising model are presented. The first, based on the treatment of Connolly and Williams for the study of phase stability in bulk alloys, consists in fitting the model parameters to the calculated internal energies of a sufficiently large number of composition profiles. The second, the generalized perturbation method (GPM), also first proposed in the context of bulk phase diagrams, derives them from the electronic structure of a reference system, relative to which the free energy of the actual configuration is expanded in powers of local concentration fluctuations. A detailed derivation of the GPM with respect to an arbitrary composition profile is given, in both the multiple-scattering and the tight-binding LMTO formalism, and recent developments of the method are discussed. Finally, the results of applications of the two approaches by different groups are presented.

Equilibrium concentration profiles of physically end tethered polystyrene molecules at the air-polymer interface

Deuteriopolystyrene of molecular weight ca 32 000 g mol −1 has been functionalized at one end with a fluorocarbon. On mixing this polymer with an unfunctionalized hydrogenous polystyrene a surface excess layer of the deuteriopolymer forms on annealing at 423 K. The surface excess as a function of deuteriopolymer content has been obtained using nuclear reaction analysis and neutron reflectometry, and the shape of the profile from neutron reflectometry alone. From previous secondary ion mass spectral data and by comparison with predictions of surface enrichment theory, it is concluded that the deuteriopolystyrene is tethered by the fluorine label to the air surface. The parameters of this polymer ‘brush’ obtained experimentally are the surface volume fraction of labelled polymer, the brush height, the surface excess and the layer thickness. The surface excess and surface volume fraction of the deuteriopolymer and the adsorption isotherm (surface excess at a function of equilibrium bulk volume fraction) have been compared to the predictions of a self-consistent field theory. Best agreement is obtained with a sticking energy of 1.6 k B T; however, there appear to be some disparities when compared to the limiting ‘dry’ brush predictions. There is evidence from neutron reflectometry of brush formation before annealing of the polymer films.

Equilibrium concentration profiles of species in CO 2—alkanolamine—water systems

Equilibrium concentrations of species and gas loading in aqueous solutions of alkanolamine loaded with CO 2 are being compared with predicted profiles obtained from the Deshmukh-Mather Model. A new technique is being proposed based on titration using a base, NaOH, to determine the concentrations of the different species at equilibrium. The technique is reliable and easy to perform to give reproducible results. Experimental and predicted concentrations of species are in good agreement over a range of gas loading between 0.4 and 1.0 for aqueous AMP solutions. For DEA, the predicted and measured values complement each other at high loading typically above 0.6. At low loading, there is a significant difference between the sets of values for carbamate and bicarbonate. These differences are likely due to the value of the equilibrium constant for the carbamate formation, which is taken as an adjustable parameter in the model, to give the best fit to the experimental data of either CO 2 partial pressure or gas loading.

Size-mismatch effect on the surface segregation of AuCu 3 alloys

A microscopic electronic theory is presented for the study of surface segregation of transition metal alloys. The total energy is written as a sum of attractive band structure energy and repulsive Born-Mayer-type interaction. The size-mismatch effect is treated within this electronic model instead of being introduced as a parameter. Through minimization of the surface free energy, the equilibrium concentration profile and surface relaxation of different faces of AuCu 3 are obtained as a function of temperature and compared with experimental data by low energy ion scattering (LEIS).

Sedimentation equilibrium of suspensions of colloidal particles at finite concentrations

Equilibrium concentration profiles of non-dilute colloidal suspensions are calculated by means of the Carnahan-Starling expression for the osmotic compressibility of hard sphere liquids. The profiles depend on the average volume fraction of the suspension, 〈ϕ〉, and on the field interaction parameter, λ0 (reciprocal of the Peclet number at infinite dilution). Profiles are computed for values of 〈ϕ〉 and λ0 typical of those encountered in sedimentation field-flow fractionation experiments. It is found that, in most cases, the volume fraction at the depletion wall is negligibly small and that the volume fraction at the accumulation wall φ0, depends on the ratio 0/λ0 only. An inflexion point is found in the concentration profile if φ0 is larger than 0.13 whatever the value of λ0.

Polymer melt near a solid wall

We develop a theory for the equilibrium concentration profile formed by a compressible polymer melt near a solid wall (or the mathematically equivalent incompressible polymer solution near a solid wall). The theory uses a Landau–Ginzburg free energy functional with a concentration dependent square gradient coefficient and an additional local contribution characterizing the influence of the wall. We introduce a mean-field algorithm for constructing the surface free energy contribution from the expression for the bulk free energy of lattice polymers. This algorithm automatically includes both energetic and entropic contributions with no adjustable parameters for lattice systems and can be applied for branched polymers as well. Approximate analytical solutions are provided for one-phase polymer density profiles at neutral, repulsive, and attractive walls. The approximate solutions reflect the behavior of the numerical solutions and display (only at lower polymer densities) a near wall linear variation of the polymer density which crosses over to an exponential approach to the bulk concentration. The numerically evaluated profiles for both athermal and nonathermal melts compare well with available Monte Carlo simulation data for a neutral wall. Physical arguments are presented which anticipate the existence of deviations between theory and simulations at higher densities. The use of lattice cluster theory free energy functions enables us for the first time to investigate the dependence of the density profile on the polymer architecture.

Compositional elastic domains in epitaxial layers of phase-separating semiconductor alloys

Abstract The global stability of a homogeneous III-V semiconductor alloy in an epitaxial film on the [001] substrate is examined with respect to an arbitrary modulation of alloy composition. The coherent phase separation in epitaxial films of unstable alloys is shown to result in the formation of structures with periodic modulations of alloy composition along the [100] (or [010]) direction of the lowest stillness in the substrate plane. The equilibrium concentration profile is governed by the interplay of the chemical and the elastic energies. The amplitude of the composition modulation is found to be maximum at the free surface and to decay in the depth of the epitaxial film. The phase diagram of the epitaxial alloy stability is calculated for a GaAs1-c,Sb c alloy.

Retention Theory of Sedimentation Field-Flow Fractionation at Finite Concentrations

In field-flow fractionation (FFF) of concentrated suspensions, particle interactions result in deviations of the equilibrium concentration profile and velocity profile from their respective exponential and parabolic infinite dilution limits. Following the Einstein diffusion formalism, the sedimentation equilibrium concentration profile of a concentrated suspension is derived for hard spheres. The velocity profile is calculated by taking into account the concentration dependence of the viscosity. The retention factor is then found to depend on two parameters, the basic FFF parameter, as usual, and the average volume fraction of the suspension

Surface segregation of the random Madelung lattice.

Using the charge-transfer model recently introduced by Magri, Wei, and Zunger [Phys. Rev. B 42, 11 388 (1990)] to study the Madelung energy of ordered and disordered binary alloys, we calculate the classical cleavage energy for a number of surfaces of a random binary alloy ${\mathit{A}}_{\mathit{x}}$${\mathit{B}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$ on a fcc lattice. We then allow x to differ from its bulk value in the surface region and determine the equilibrium concentration profile by minimizing the total energy. We also present results for the energy necessary to achieve a given surface concentration ${\mathit{x}}_{1}$ of species A for different values of the bulk composition.

A simple quasi-3-D model of suspended sediment transport in a nonequilibrium state

A new simple model to evaluate quasi-3-D suspended sediment transport in a non-equilibrium state has been developed through a formulation based on a kind of weighted residual concept. The model includes only two unknown variables to be solved in a horizontal 2-D domain ( x, y); i.e., the sediment concentration at the bottom C b( x, y, t) and the shape factor A( x, y, t) in the assumed exponential curve for the vertical profile of the concentration, both of which are the most important factors governing the nonequilibrium sedimentary process. From the computational results for the 1-D deposition and erosion problems, the model has been confirmed to give satisfactory results in accuracy and in computational time. The present model has also been applied to the 3-D suspended sediment transport problem in the nearshore region including the surf zone. The comparison between the computational results by the present model and those by the conventional models assuming the local equilibrium concentration profile and the nonequilibrium but vertically uniform profile, have demonstrated the importance of taking the effect of the nonequilibrium 3-D condition into the consideration for such problems.

Studies on reductive back extraction of plutonium in Purex process

Kinetic equation of one stage process of reduction of plutonium(IV) in two phase 30 vol. % TBP — n-dodecane/HNO3 system is given. The expression derived is used for elaborating the method of calculation of equilibrium (ultimate) concentration profiles of plutonium in multistage contactors for countercurrent process of U/Pu separation.

Anomalous penetrant transport in glassy polymers: 4. Stresses in partially swollen polymers

The equilibrium swelling process and stress evolution of glassy polymer systems undergoing Case-II penetrant transport were examined using a rubber elasticity theory. Consideration of coupling between stresses and penetrant concentration permitted calculation of equilibrium stress and concentration profiles for a variety of molecular and geometric factors.

Adsorption on stepped surfaces: The adsorbate concentration profile and its induced work-function change.

A model for the work-function change caused by adsorption on stepped surfaces is studied and discussed. The equilibrium concentration profiles of adsorbed particles at step and terrace sites at different coverage and temperature is calculated by means of a Monte Carlo simulation. The obtained profiles allow us to evaluate the work-function changes by use of the Helmholtz equation and the experimental evidence which shows that the dipole moment of the adparticles depends on the adsorption site. Our results are in agreement with available experimental data and we point out that special care must be taken in order to interpret the work-function changes measured by use of polycrystalline samples and step-``contaminated'' single crystals.

Particles segregation in fluidized bed binary systems

Binary systems have been divided into six groups according to their minimum fluidization velocities, densities, and size of particles. A concentration profile of fluidized bed has been studied. Experimental results have indicated that interval (45 - 90 sec.) is necessary for concentration profile generation. The profile appears to be uninfluenced by original arrangement of system components, the composition of bulk and grid region being in mutual equilibrium. Superficial velocity strongly affects system equilibrium, especially at higher jetsam concentrations. The best segregation of the system was achieved at velocities close to the minimum fluidization velocity of the binary system. The binary systems of the 3rd and 6th types change from segregated to mixed fluidized bed if gas velocity increases. The binary system of the 1st type was difficult to mix up since flotsam entrained partially if velocity of air was increased. The equilibrium concentration profile was uninfluenced by the size of the equipment - provided fluidized bed height was above a critical limit.

Thermodynamic Study of Silane-hydrogen Chemical System

AbstractThe thermodynamic calculations of chemical equilibrium compositions of various species were performed for silane-hydrogen system at different temperatUres [500-6000K], pressures [0.01-10 Torr], and mixing ratios [various silane fractions]. The effect of dopant impurities such as diborane and phosphine on equilibrium concentration profiles of various gaseous, tonic, and condensed species was also investigated. These data will be very useful in understanding the deposition chemistry of silane chemical systems and will be a valuable aid in optimization studies.

Measurements of the turbulence characteristics of sand suspended by a tidal current

Measurements made with a new fast-response suspended sand sensor have for the first time enabled the turbulence characteristics of a natural sand suspension to be studied. Suspended sand concentration measurements, together with fast-response current measurements, were made 18 cm above a sandy bed under a strong tidal current. They showed a highly variable concentration field, dominated by clouds of sand which took 2 to 10 s to sweep past the sensor. The concentration spectrum had peak energy at a wavelength of about 3 m, and exhibited a −5/3 power dependence at high frequencies. Damping of the turbulence intensity of the current was observed when sand was suspended. A spectral approach produced more plausible values for the upward Reynolds flux of sediment than the direct covariance technique. The calculated upward fluxes were appreciably smaller than the settling fluxes around the time of maximum current, contrary to the expectation for an equilibrium concentration profile. The turbulence characteristics of the concentration field displayed marked similarities with standard results from atmospheric temperature measurements, strengthening assumptions commonly made in the prediction of sediment transport rates.

Molecular Properties of Lipoprotein Lipase

The monomer molecular size of bovine lipoprotein lipase was evaluated by sedimentation equilibrium measurements and by gel permeation chromatography in 6 M guanidinium chloride. To establish molecular weight unequivocally we determined the partial specific volume (v) experimentally. This was done by analyzing equilibrium concentration profiles from analytical ultracentrifugation in 6 M guanidinium chloride using buffers made up in H2O and 2H2O. The combined results gave a v of 0.71 +/- 0.007 ml/g and a molecular weight of 41,700 +/- 1000 for monomeric bovine lipoprotein lipase. This value did not change upon mild tryptic digestion; the elution volume upon gel permeation chromatography in 6 M guanidinium chloride was also unaffected by treatment with trypsin. Sedimentation equilibrium measurements of the trypsin-treated material in the presence of reducing agents gave limiting molecular weights of 19,000 and 23,000, demonstrating that mild trypsin digestion cleaved lipoprotein lipase into two polypeptide chains of similar size held together by disulfide bonds. Mild trypsin digestion also resulted in a loss of secondary structure as determined by circular dichroic measurements. Discussion centers around the correlation between these effects of trypsin on the molecular properties of lipoprotein lipase and the previously reported effects on the kinetic properties of the enzyme.

Equilibrium thermodynamic correlations in coal hydrogenation to pipeline gas

A Fortran IV computer program was developed to calculate the composition at equilibrium of products from a single-stage coal gasification process involving coal, H2 and steam. This program was used to develop equilibrium concentration profiles for various components and heating values of dry product gas as a function of temperature, pressure and percentage steam in a hydrogen-steam mixture feed gas. Based on the results, a useful range for the variables, from the thermodynamic point of view, has been established.