Dilute Binary Mixtures Research Articles

Interfacial wave theory of dendritic growth from a binary mixture: A comparison with experiments

This paper presents a wave theory for dendritic growth from a dilute binary mixture and its comparison with experiments. Our study shows that the global instability mechanism, the so-called global trapped wave (GTW) mechanism, that we first explored for dendritic solidification from a pure substance is valid for the present systems in a very similar fashion. Due to the presence of this instability mechanism, the present system also permits a discrete set of unstable global modes, which we call the global trapped wave (GTW) modes, and a unique global neutrally stable (GNS) mode. This GNS mode solution is related to the material parameters and the growth conditions. It determines the micro-structure of the dendrite and selects its tip-velocity. We have completed the numerical computations of the GNS mode solution for a large range of various physical parameters, and compared the theoretical results with the available experimental data, without any adjustable parameters. The theory shows good agreement with the experiments.

Transient effect on a catalytic reactor

At time t < 0, a steady motion of a chemically reacting dilute binary mixture of a gas exists in a long cylindrical porous pipe, such that the steady-state velocity component and the gas concentration are given by (us, vs, O) and cs in cylindrical polar co-ordinates (r, o, z). When t = O, a constant pressure gradient is applied at one end of the tube. If the inertia terms are neglected, a Laplace transform and a finite-difference scheme are employed to solve for the transient flow (u, v. w) and c. The time required to achieve a steady state is different for w on one hand, and for u, v and c on the other hand.

Long-wavelength analysis of the coupling between solidification and Soret-driven convection at positive separation ratio.

An investigation is made of the coupling between Soret-driven convection, at positive separation ratio, and a solid-liquid interface. This phase boundary forms upon the solidification of an upper portion of a thin layer of a dilute binary mixture that is confined between highly conducting and nearly impermeable horizontal plates. Using a long-wavelength expansion, a nonlinear evolution equation is derived that includes the coupled effects of solidification and thermosolutal convection due to the Soret effect. The validity of the derivation is found to be limited to regions, in the parameter space, that are determined as functions of the separation ratio, the Lewis number, and the thickness of the solid layer. The numerical solutions of the evolution equation show that the formation of the solid layer affects the leading-order concentration profile, and that the deviation of the interface from its planar state is weakly affected by the convective currents in the liquid. The interface morphology, however, has the same form as the convective pattern in the fluid mixture. The consideration of boundaries that are poor conductors of heat leads to a problem that is characterized by a heat-transfer Biot number \ensuremath{\gamma} and a mass-transfer Biot number \ensuremath{\beta}. The threshold values for the onset of convection are determined, and their dependence on \ensuremath{\beta} and \ensuremath{\gamma} is discussed.

Bifurcation Analysis of Cellular Interfaces in Unidirectional Solidification of a Dilute Binary Mixture

Analytical bifurcation theory is used to study interface patterns in the unidirectional solidification of a dilute binary mixture. Armbruster and Dangelmayr’s classification of bifurcation equations equivariant under the diagonal action in $\mathbb{C}^2 $ of the symmetry group $O( 2 )$ is used to show that the local bifurcation structure of the model long-wave evolution equation derived by Riley and Davis is equivalent to that of a codimension-one normal form. xamination of the bifurcation structure of the normal form then leads to a description of the development of three-dimensional interface patterns during unidirectional solidification. The general methodology underlying this study is also discussed, and conditions under which the method can be applied to other evolution equations are derived and clearly stated.

Directional solidification with interface dissipation.

The front dynamics during directional solidification of a dilute binary mixture with noninstantaneous interface kinetic attachment is investigated in the weakly and highly nonlinear regimes. In the linear regime we find that, even for small enough kinetic coefficient, the most unstable mode is appreciably modified. With the help of multiscale analysis, we derive the amplitude equation that governs the front dynamics close to criticality. It is found that, for the case of a constant miscibility gap, studied here, the supercritical nature of the bifurcation remains unaltered. However, the computation of the Landau coefficient shows that the interface excursion is reduced by kinetics. Moreover, the analysis of the Mullins-Sekerka spectrum close to the instability threshold indicates that higher harmonics should be activated by interface kinetics. This is confirmed numerically. We study the transition from cells to dendrites by adopting a recent code to the present situation. We find scaling laws that confirm those obtained in the free-growth situation [Brener and Mel'nikov (unpublished)]. The interface structure is ``angular,'' and side branches are much less pronounced, even when the P\'eclet number exceeds a few unities, than those obtained in the free-kinetics case [Saito, Misbah, and M\uller-Krumbhaar, Phys. Rev. Lett. 63, 2377 (1989)]. Our findings agree well with experiments [P. Kurowski, C. Guthmann, and S. de Cheveign\'e (unpublished)] on impure ${\mathrm{CBr}}_{4}$. We speculate that dendrites in this system are selected by kinetic anisotropy rather than by surface-tension anisotropy.

Bifurcation analysis of solutal convection during directional solidification

Techniques of bifurcation theory are used to analyze the possible forms of steady, two-dimensional, buoyancy-driven flow present during the vertical directional solidification of a dilute binary mixture. Morphological instabilities of the crystal–melt interface are ignored and the interface is assumed to be planar. The focus of the study concerns the effect of lateral confinement on the flow patterns observable in this simple model system. The use of a time-dependent numerical scheme by McFadden and Coriell [Proceedings of the First National Fluid Dynamics Congress, Cincinnati (AIAA, Washington, DC, 1988), p. 1572] to investigate this same problem resulted in gaps in the bifurcation structure, because unstable steady states exist that cannot be computed by their method. The present work extends and completes the description of the possible steady states. In particular, the complex modal exchanges that occur as the separation between the lateral walls varies are elucidated, and stable flow regimes and the sensitivity of the results to the closure condition at the upper boundary are determined.

Directional Solidification of a Faceted Crystal

When a dilute binary mixture is pulled at constant velocity V in an external thermal gradient G, the solid-liquid interface exhibits a morphological instability when V becomes larger than a threshold velocity Vc(G). For V Vc it develops a (quasi)periodic cellular structure. This instability, first analyzed by Mullins and Sekerka(1), results from the competition between the destabilizing effect of solute diffusion and the stabilization due to the thermal gradient and to capillarity.

Long-Wave Morphological Instabilities in the Directional Solidification of a Dilute Binary Mixture

Long-wave instabilities in a directionally-solidified binary mixture may occur in several limits. Sivashinsky identified a small-segregation-coefficient limit and obtained a weakly-nonlinear evolution equation governing subcritical two-dimensional bifurcation. Brattkus and Davis identified a near-absolute-stability limit and obtained a strongly-nonlinear evolution equation governing supercritical two-dimensional bifurcation. The present investigation identifies a third, strongly-nonlinear, evolution equation, arising in the small-segregation-coefficient, large-surface-energy limit. This equation links both of the former and describes the change from the sub- to super-critical bifurcations. This study sets the previous long-wave analyses into a logical framework.

Experimental Study of Surfactant Effects on Pool Boiling Heat Transfer

In the first part of this work, nucleate boiling of aqueous solutions of sodium lauryl sulfate (SLS) over relatively wide ranges of concentration and heat flux was carried out in a pool boiling apparatus. The experimental results show that a small amount of surface active additive makes the nucleate boiling heat transfer coefficient h considerably higher, and that there is an optimum additive concentration for higher heat fluxes. Beyond this optimum point, further increase in additive concentration makes h lower. In the second part of this work, nucleate boiling heat transfer rate for n-propanol-water binary mixtures with various amounts of sodium lauryl sulfate were measured in the same pool boiling apparatus. The importance of the mass diffusion effect, which is caused by preferential evaporation of the more volatile component at the vapor-liquid interface on the boiling of the binary mixture, has been confirmed. However, it is shown that the effect exerted by the addition of a surfactant dominates over the mass diffusion effect in dilute binary mixtures.

Kinetic-sound propagation in dilute gas mixtures.

Kinetic sound is predicted in dilute disparate-mass binary gas mixtures, propagating exclusively in the light compound and much faster than ordinary sound. It should be detectable by light-scattering experiments, as an extended shoulder in the scattering cross section for large frequencies. As an example, H/sub 2/-Ar mixtures are discussed.

Determination of binary gas diffusion coefficients in spherical porous media by steady and unsteady-state analysis of single pellet reactor data

A mathematical model that describes both the transient and steady-state response of a single spherical porous catalyst pellet to a step input in concentration of a dilute binary gas mixture is presented. The model equations, which contain the effective diffusion coefficient of the dilute binary as the only parameter, are reduced to a set of triple-series equations that are solved using the least square method of weighted residuals. To test the theory, an experimental apparatus was constructed and used to collect both transient response and steady-state data for a helium-nitrogen binary gas pair. Effective diffusion coefficients determined from these measurements using time-domain and Laplace transform domain parameter estimation are shown to be within the correct order-of-magnitude and agree within 25% of each other. Some possible extensions of the model are also suggested.

The effect of molecular polarizability on the properties of fluids

We report a detailed study of the effect of polarizability (both isotropic and anisotropic) on the properties of dilute fluids and binary fluid mixtures. The properties studied include the second pressure and dielectric virial coefficients, the minimum pair interaction energy and structure, the radial distribution function and the structure factor. Our results show that polarizability makes a significant contribution to all the properties studied. The effect of polarizability becomes more important when mixture components have quadrupole moments of opposite signs. Anisotropy of polarizability in general is less important, except in the case of the second dielectric virial coefficient, or when the two molecules have quadrupoles of opposite signs. From the magnitude of these contributions, it also appears unlikely that effective models for dilute fluids can account for polarizability adequately, when it is not explicitly included in the potential model.

Influence of alkali cations on the infrared spectra of water molecules in aprotic solvents

The influence of alkali cations on the frequency of the OH vibration of H-bonded HOD (or H/sub 2/O) in dilute ternary mixtures (salt/water/aprotic solvent with H-bond acceptor groups B) is observed. The authors find that the magnitude of the frequency shift due to the cation increases proportionally with the frequency shift due to the O-H...B H bond in the dilute binary (water/solvent) mixture. The correlation between OH-frequency shifts in different solvents (OH...B complexes) and the additional shift due to Li/sup +/, Na/sup +/, and K/sup +/ salts is linear. The slopes of these correlations may be taken as a measure of the cooperative influence of the cations. The slopes are Li/sup +/, 0.55; Na/sup +/, 0.40; K/sup +/, approx. 0.08. Some possible consequences of this cooperativity on H-bond energies, enthalpies of solutions, and O-O distances in aqueous salt solutions are discussed.

Second order in the gradients effects in a dilute binary mixture

A study is made to compute relevant transport properties for a dilute binary mixture of inert gases to second order in the gradients without explicitly solving the Boltzmann equation to that order. This is done with the Chapman-Enskog method, seeking to express such quantities in terms of the solution to first order. The pressure tensor and the velocity of diffusion are two quantities which allow for this computation. In the particular case when the sum of the particle densities of the mixture (n A, n B) is constant, one finds that in order to keep the Chapman-Enskog method mathematically consistent, it is necessary that the divergence of the mass velocity be position independent. Finally, we consider the case of swarms of charged particles and study the prediction of the method in the Navier-Stokes and Burnett regimes for diffusion phenomena. In the latter case, the results are restricted to electrons in a gas.

Forced modes of dilute binary gas mixtures

Anomalous forced sound dispersion in dilute binary gas mixtures is studied as a function of the mass ratio of the two components, using one- and two-temperature theories as well as different interparticle potentials. For a disparatemass mixture, such as He-Xe, the results are compared with previous work due to Johnson et al. It is suggested that even for nondisparate-mass mixtures, a one-temperature treatment is not appropriate.

Sorption and pervaporation of dilute aqueous solutions of organic compounds through polymer membranes

Pervaporation of dilute aqueous binary mixtures of four organic compounds (benzene, chloroform, acetone and ethanol) through nitrile—butadiene and styrene—butadiene copolymers was investigated. A pervaporation device has been built, which allows measurement of the pervaporation flux and selectivity of a membrane as a function of the upstream composition of the feed and the downstream total pressure of the pervaporate. In order to relate pervaporation results to equilibrium properties of the membranes, the sorption of water and dilute aqueous solutions was mainly investigated. The pervaporation of dilute aqueous solutions of benzene and chloroform has been extensively studied, including the separation of traces of chloroform, and is modelled through a “sixcoefficients exponential model” [1]. This model is derived from a solution—diffusion analysis of the selective transfer, assuming an exponential dependence of both diffusivities on concentrations of both permeants. Semi-quantitative information about the potential interactions existing in the system solute i—solvent j—membrane and about the concentration profiles at steady-state may be derived from these coefficients.

Excess second virial coefficients for some dilute binary gas mixtures

Excess and interaction second virial coefficients have been measured in the temperature range 290-320 K for 20 binary gaseous systems. Several of these systems have previously been studied by Brewer. The present values are in excellent agreement with his results. The excess virial coefficients have been used to calculate excess molar enthalpies of mixing at 300 K.

The effect of quasispherical and chainlike solutes on the nematic to isotropic phase transition in liquid crystals

The effects of solute molecular structure (size, shape and flexibility) and solvent molecular structure (length-to-breadth ratio and end-chain flexibility) on nematic phase stability in dilute binary mixtures of nonmesomorphic solutes and nematogenic solvents are examined through experiment and theory. Addition of the perturbing solute to the liquid-crystalline solvent leads to depression of the nematic–isotropic (NI) transition temperature and formation of a two-phase region. Directly determined moduli of the slopes, βn and βi, of the nematic and isotropic phase boundary lines in the reduced NI transition temperature (T*) vs solute mole fraction (x2) diagrams are reported for quasispherical and chainlike solutes with two nematogenic solvents. The systems studied are the quasispheres Et4C (tetraethylmethane) and R4Sn (R = CH3, C2H5, n-C3H7 and n-C4H9) and the chains n-C8H18 through n-C14H30, mixed with p-methoxybenzylidine-p′-n-butylaniline (MBBA) and p-n-pentyl-p′-cyanobiphenyl (5CB). Also reported are indirectly determined β∞n and β∞i values (limit as x2→0), using a novel approach combining differential scanning calorimetry (for the pure solvent contribution) and gas–liquid chromatography (for the solution contribution), for Et4C and n-C5H12 through n-C11H24, with MBBA, 5CB, p-azoxyanisole (PAA), and p,p′-di-n-hexyloxyazoxybenzene (DHAB). For the systems in common, the average difference between the directly and indirectly determined β values is approximately 10% and, qualitatively consistent with lattice model predictions, the comparison suggests slight curvature of the phase boundary lines. The experimental β values, as a function of increasing solute size, are found to double (roughly) for the quasispheres and increase only slightly for the chains, reflecting the concurrent behavior of the solution contribution to β. The thermodynamic results for the quasispherical solutes are compared in some detail with predicted values from statistical–mechanical theories based on rigid-rod solvent molecules: (a) lattice model, (b) virial expansion treatment, (c) molecular-field model (after Maier and Saupe), and (d) van der Waals model. All four models correctly predict the observed trend of increasing βn and βi with increasing solute size and yield predicted slopes which are within a factor of 2 of experiment. All are found to be deficient to a minor or major extent in their predictions of the solvent and solution contributions to the β values. The more tractable lattice model is used to examine the chainlike solutes and the effect of solvent end-chain flexibility. It correctly predicts the qualitative features of the observed dependence of β on solute size for the different solute structures (including rigid-rod solutes) and indicates that dissolved n-alkane solutes have appreciable (effective) chain flexibility in nematic solvents. Incorporation of some solvent end-chain flexibility in the lattice model markedly improves agreement with experiment, primarily through better quantitative prediction of the solution contribution.

Second order kinetic properties of dilute binary noble gas mixtures containing helium

Second order kinetic properties of dilute binary noble gas mixtures containing helium

The empirical Kestin model and the second order transport properties of binary mixtures of Ne, Ar, Kr and Xe

It is showed from the experimental αT data for the dilute binary mixtures of Ne, Ar, Kr and Xe at 340 K that the law of corresponding states proposed by Kestin, Ro and Wakeham is able to predict the mentioned second order kinetic property altogether with the first order transport and thermodynamic properties of the examined mixtures.