Density change effects on crystal growth from the melt

Phase-field modelling of the effect of density change on solidification revisited: model development and analytical solutions for single component materials

In this paper the development of a physically consistent phase-field theory of solidification shrinkage is presented. The coarse-grained hydrodynamic equations are derived directly from the N-body Hamiltonian equations in the framework of statistical physics, while the constitutive relations are developed in the framework of the standard phase-field theory, by following the variational formalism and the principles of non-equilibrium thermodynamics. To enhance the numerical practicality of the model, quasi-incompressible hydrodynamic equations are derived, where sound waves are absent (but density change is still possible), and therefore the time scale of solidification is accessible in numerical simulations. The model development is followed by a comprehensive mathematical analysis of the equilibrium and propagating one-dimensional solid-liquid interfaces for different density-phase couplings. It is shown, that the fluid flow decelerates/accelerates the solidification front in case of shrinkage/expansion of the solid compared to the case when no density contrast is present between the phases. Furthermore, such a free energy construction is proposed, in which the shape of the equilibrium planar phase-field interface is independent from the density-phase coupling, and the equilibrium interface represents an exact propagating planar interface solution of the quasi-incompressible hydrodynamic equations. Our results are in agreement with previous theoretical predictions.

Phase-field modeling of isothermal quasi-incompressible multicomponent liquids

In this paper general dynamic equations describing the time evolution of isothermal quasi-incompressible multicomponent liquids are derived in the framework of the classical Ginzburg-Landau theory of first order phase transformations. Based on the fundamental equations of continuum mechanics, a general convection-diffusion dynamics is set up first for compressible liquids. The constitutive relations for the diffusion fluxes and the capillary stress are determined in the framework of gradient theories. Next the general definition of incompressibility is given, which is taken into account in the derivation by using the Lagrange multiplier method. To validate the theory, the dynamic equations are solved numerically for the quaternary quasi-incompressible Cahn-Hilliard system. It is demonstrated that variable density (i) has no effect on equilibrium (in case of a suitably constructed free energy functional) and (ii) can influence nonequilibrium pattern formation significantly.

Consequences of CO2 solubility for hydrate formation from carbon dioxide containing water and other impurities

Deciding on the upper bound of water content permissible in a stream of dense carbon dioxide under pipeline transport conditions without facing the risks of hydrate formation is a complex issue. In this work, we outline and analyze ten primary routes of hydrate formation inside a rusty pipeline, with hydrogen sulfide, methane, argon, and nitrogen as additional impurities. A comprehensive treatment of equilibrium absolute thermodynamics as applied to multiple hydrate phase transitions is provided. We also discuss in detail the implications of the Gibbs phase rule that make it necessary to consider non-equilibrium thermodynamics. The analysis of hydrate formation risk has been revised for the dominant routes, including the one traditionally considered in industrial practice and hydrate calculators. The application of absolute thermodynamics with parameters derived from atomistic simulations leads to several important conclusions regarding the impact of hydrogen sulfide. When present at studied concentrations below 5 mol%, the presence of hydrogen sulfide will only support the carbon-dioxide-dominated hydrate formation on the phase interface between liquid water and hydrate formers entering from the carbon dioxide phase. This is in contrast to a homogeneous hydrate nucleation and growth inside the aqueous solution bulk. Our case studies indicate that hydrogen sulfide at higher than 0.1 mol% concentration in carbon dioxide can lead to growth of multiple hydrate phases immediately adjacent to the adsorbed water layers. We conclude that hydrate formation via water adsorption on rusty pipeline walls will be the dominant contributor to the hydrate formation risk, with initial concentration of hydrogen sulfide being the critical factor.

Phase field modelling of spinodal decomposition in the oil/water/asphaltene system

In this paper the quantitative applicability of van der Sman/van der Graaf type Ginzburg–Landau theories of surfactant assisted phase separation [van der Smanet al.,Rheol. Acta, 2006,46, 3] is studied for real systems displaying high surfactant concentrations at the liquid–liquid interface.

Unified derivation of phase-field models for alloy solidification from a grand-potential functional

In the literature, two quite different phase-field formulations for the problem of alloy solidification can be found. In the first, the material in the diffuse interfaces is assumed to be in an intermediate state between solid and liquid, with a unique local composition. In the second, the interface is seen as a mixture of two phases that each retain their macroscopic properties, and a separate concentration field for each phase is introduced. It is shown here that both types of models can be obtained by the standard variational procedure if a grand-potential functional is used as a starting point instead of a free energy functional. The dynamical variable is then the chemical potential instead of the composition. In this framework, a complete analogy with phase-field models for the solidification of a pure substance can be established. This analogy is then exploited to formulate quantitative phase-field models for alloys with arbitrary phase diagrams. The precision of the method is illustrated by numerical simulations with varying interface thickness.

Pressure effects for crystal growth in a closed system

We analyze the growth of a crystal from its supercooled liquid in a closed domain (constrained growth), taking into account the effects due to the different densities rho(s) and rho(l) of the solid and liquid phases. We assume rho(l) > rho(s), i.e., the liquid expands upon solidification. Then, the growth is contrasted by an increasing pressure, which results in a continuous decrease of the coexistence temperature and the effective supercooling. These phenomena have been simulated in two dimensions through a modified version of the classic phase-field model. We observe that for spherical growth the interface temperature reflects almost instantaneously the change of the coexistence temperature. For dendritic growth, we observed a relaxation time for the dendrite tip velocity and the tip radius which is comparable to the characteristic time of the process; however, after the first fast transient, the growth dynamics seems to follow the changing pressure with no appreciable lag. The onset of the morphological instability is slightly anticipated in respect to free growth.

Interface dynamics and solute trapping in alloy solidification with density change

We present a phase-field model for the solidification of a binary alloy, which incorporates hydrodynamic effects due to the different densities of the solid and liquid phases. We start from a generalized thermodynamic potential with squared gradient terms for the associated fields; the condition of local positive entropy production is then utilized to derive a set of equations that drive the system towards equilibrium. The model has been numerically solved in one dimension, to investigate the effects of the flow field on the interface dynamics. We observed that solute trapping is almost unaffected by the fluid advection, while the interface mobility is strongly reduced as the fluid velocity increases. This reflects on the dependence of the interface temperature T(I) on the growth rate v(I): the region of the unstable (ascending) branch is reduced, and the maximum of the T(I)(v(I)) curve is shifted towards lower velocities.

Kinetics of growth process controlled by convective fluctuations

A model of the spherical (compact) growth process controlled by a fluctuating local convective velocity field of the fluid particles is introduced. It is assumed that the particle velocity fluctuations are purely noisy, Gaussian, of zero mean, and of various correlations: Dirac delta, exponential, and algebraic (power law). It is shown that for a large class of the velocity fluctuations, the long-time asymptotics of the growth kinetics is universal (i.e., it does not depend on the details of the statistics of fluctuations) and displays the power-law time dependence with the classical exponent 1/2 resembling the diffusion limited growth. For very slow decay of algebraic correlations of fluctuations asymptotically like t(-gamma), gamma in (0,1]), kinetics is anomalous and depends strongly on the exponent gamma. For the averaged radius of the crystal <R(t)> approximately t(1-gamma/2) for 0<gamma<1 or <R(t)> approximately (t ln t)1/2 for gamma=1.