Amplitude equations for systems with long-range interactions

We derive amplitude equations for interface dynamics in pattern forming systems with long-range interactions. The basic condition for the applicability of the method developed here is that the bulk equations are linear and solvable by integral transforms. We arrive at the interface equation via long-wave asymptotics. As an example, we treat the Grinfeld instability, and we also give a result for the Saffman-Taylor instability. It turns out that the long-range interaction survives the long-wave limit and shows up in the final equation as a nonlocal and nonlinear term, a feature that to our knowledge is not shared by any other known long-wave equation. The form of this particular equation will then allow us to draw conclusions regarding the universal dynamics of systems in which nonlocal effects persist at the level of the amplitude description.Comment: LaTeX source, 12 pages, 4 figures, accepted for Physical Review

Phase Field Modeling of Fast Crack Propagation

We present a continuum theory which predicts the steady state propagation of cracks. The theory overcomes the usual problem of a finite time cusp singularity of the Grinfeld instability by the inclusion of elastodynamic effects which restore selection of the steady state tip radius and velocity. We developed a phase field model for elastically induced phase transitions; in the limit of small or vanishing elastic coefficients in the new phase, fracture can be studied. The simulations confirm analytical predictions for fast crack propagation.Comment: 5 pages, 11 figure

Experimental And Theoretical Study Of The Sign Preference In The Nucleation Of Water Vapor

The nucleation of water vapor on ions in atmospheres of helium and argon was studied using an expansion type cloud chamber. Separation of the positive and negative ions was achieved so that the nucleation could be studied as a function of both the sign of the ionic charge and the supersaturation. A semiphenominological theory was developed as an extension of the classical liquid drop theory to include the effects of the ionic charge on the nucleation process. The theoretical model of the prenucleation embryo was assumed to possess an oriented dipole surface layer with the direction of orientation dependent on the sign of the ionic charge. The theory predicts not only the increase in the nucleation rate compared to that for homogeneous nucleation and a difference in rate between positive and negative ions, the negative ions having the higher nucleation rate, but also predicts a correction term to the classical theory of homogeneous nucleation for polar molecules which exhibit an electrical double layer at the liquid surface. Comparison of the theoretical and experimental results for nucleation on both positive and negative ions yields good agreement and indicates the prenucleation embryo is probably a tightly bonded highly structured cluster possessing an oriented dipole surface layer. © 1971

A Correction To Classical Homogeneous Nucleation Theory For Polar Molecules Exhibiting An Electric Double Layer At The Liquid Surface

An oriented dipole surface layer is added to the classical liquid drop model of nucleation to account for the surface behavior of substances having polar nonsymmetrical molecules. The preliminary treatment of the change in free energy as given by Abraham is modified to include Fletcher\u27s exponential decay of the degree of orientation. An additional correction for the excess binding energy due to the presence of a foreign molecule in the prenucleation cluster is included to account for the inflections observed in the experimental results of Allen and Kassner. The resulting free energy of formation is combined with the kinetic treatment of Frenkel to obtain a nucleation rate law. The theoretical results are compared to the experimental results of Allen and Kassner as a function of both supersaturation and temperature. The agreement is good once the heterogeneous component is taken into account. © 1972

Influence of uniaxial stress on the lamellar spacing of eutectics

Directional solidification of lamellar eutectic structures submitted to uniaxial stress is investigated. In the spirit of an approximation first used by Jackson and Hunt, we calculate the stress tensor for a two-dimensional crystal with triangular surface, using a Fourier expansion of the Airy function. crystal with triangular surface in contact with its melt, given that a uniaxial external stress is applied. The effect of the resulting change in chemical potential is introduced into the standard model for directional solidification of a lamellar eutectic. This calculation is motivated by an observation, made recently [I. Cantat, K. Kassner, C. Misbah, and H. M\"uller-Krumbhaar, Phys. Rev. E, in press] that the thermal gradient produces similar effects as a strong gravitational field in the case of dilute-alloy solidification. Therefore, the coupling between the Grinfeld and the Mullins-Sekerka instabilities becomes strong, as the critical wavelength of the former instability gets reduced to a value close to that of the latter. Analogously, in the case of eutectics, the characteristic length scale of the Grinfeld instability should be reduced to a size not extremely far from typical lamellar spacings. In a Jackson-Hunt like approach we average the undercooling, including the stress term, over a pair of lamellae. Following Jackson and Hunt, we assume the selected wavelength to be determined by the minimum undercooling criterion and compute its shift due to the external stress. we realize the shifting of the wavelength by the application of external stress. In addition, we find that in general the volume fraction of the two solid phases is changed by uniaxial stress. Implications for experiments on eutectics are discussed.Comment: 8 pages RevTex, 6 included ps-figures, accepted for Phys. Rev.

Fracture in Mode I using a Conserved Phase-Field Model

We present a continuum phase-field model of crack propagation. It includes a phase-field that is proportional to the mass density and a displacement field that is governed by linear elastic theory. Generic macroscopic crack growth laws emerge naturally from this model. In contrast to classical continuum fracture mechanics simulations, our model avoids numerical front tracking. The added phase-field smoothes the sharp interface, enabling us to use equations of motion for the material (grounded in basic physical principles) rather than for the interface (which often are deduced from complicated theories or empirical observations). The interface dynamics thus emerges naturally. In this paper, we look at stationary solutions of the model, mode I fracture, and also discuss numerical issues. We find that the Griffith's threshold underestimates the critical value at which our system fractures due to long wavelength modes excited by the fracture process.Comment: 10 pages, 5 figures (eps). Added 2 figures and some text. Removed one section (and a figure). To be published in PR

Pattern formation in directional solidification under shear flow. I: Linear stability analysis and basic patterns

An asymptotic interface equation for directional solidification near the absolute stabiliy limit is extended by a nonlocal term describing a shear flow parallel to the interface. In the long-wave limit considered, the flow acts destabilizing on a planar interface. Moreover, linear stability analysis suggests that the morphology diagram is modified by the flow near the onset of the Mullins-Sekerka instability. Via numerical analysis, the bifurcation structure of the system is shown to change. Besides the known hexagonal cells, structures consisting of stripes arise. Due to its symmetry-breaking properties, the flow term induces a lateral drift of the whole pattern, once the instability has become active. The drift velocity is measured numerically and described analytically in the framework of a linear analysis. At large flow strength, the linear description breaks down, which is accompanied by a transition to flow-dominated morphologies, described in a companion paper. Small and intermediate flows lead to increased order in the lattice structure of the pattern, facilitating the elimination of defects. Locally oscillating structures appear closer to the instability threshold with flow than without.Comment: 20 pages, Latex, accepted for Physical Review

A Supercooled Spin Liquid State in the Frustrated Pyrochlore Dy2Ti2O7

A "supercooled" liquid develops when a fluid does not crystallize upon cooling below its ordering temperature. Instead, the microscopic relaxation times diverge so rapidly that, upon further cooling, equilibration eventually becomes impossible and glass formation occurs. Classic supercooled liquids exhibit specific identifiers including microscopic relaxation times diverging on a Vogel-Tammann-Fulcher (VTF) trajectory, a Havriliak-Negami (HN) form for the dielectric function, and a general Kohlrausch-Williams-Watts (KWW) form for time-domain relaxation. Recently, the pyrochlore Dy2Ti2O7 has become of interest because its frustrated magnetic interactions may, in theory, lead to highly exotic magnetic fluids. However, its true magnetic state at low temperatures has proven very difficult to identify unambiguously. Here we introduce high-precision, boundary-free magnetization transport techniques based upon toroidal geometries and gain a fundamentally new understanding of the time- and frequency-dependent magnetization dynamics of Dy2Ti2O7. We demonstrate a virtually universal HN form for the magnetic susceptibility, a general KWW form for the real-time magnetic relaxation, and a divergence of the microscopic magnetic relaxation rates with precisely the VTF trajectory. Low temperature Dy2Ti2O7 therefore exhibits the characteristics of a supercooled magnetic liquid; the consequent implication is that this translationally invariant lattice of strongly correlated spins is evolving towards an unprecedented magnetic glass state, perhaps due to many-body localization of spin.Comment: Version 2 updates: added legend for data in Figures 4A and 4B; corrected equation reference in caption for Figure 4

Influence of external flows on crystal growth: numerical investigation

We use a combined phase-field/lattice-Boltzmann scheme [D. Medvedev, K. Kassner, Phys. Rev. E {\bf 72}, 056703 (2005)] to simulate non-facetted crystal growth from an undercooled melt in external flows. Selected growth parameters are determined numerically. For growth patterns at moderate to high undercooling and relatively large anisotropy, the values of the tip radius and selection parameter plotted as a function of the Peclet number fall approximately on single curves. Hence, it may be argued that a parallel flow changes the selected tip radius and growth velocity solely by modifying (increasing) the Peclet number. This has interesting implications for the availability of current selection theories as predictors of growth characteristics under flow. At smaller anisotropy, a modification of the morphology diagram in the plane undercooling versus anisotropy is observed. The transition line from dendrites to doublons is shifted in favour of dendritic patterns, which become faster than doublons as the flow speed is increased, thus rendering the basin of attraction of dendritic structures larger. For small anisotropy and Prandtl number, we find oscillations of the tip velocity in the presence of flow. On increasing the fluid viscosity or decreasing the flow velocity, we observe a reduction in the amplitude of these oscillations.Comment: 10 pages, 7 figures, accepted for Physical Review E; size of some images had to be substantially reduced in comparison to original, resulting in low qualit

Pattern Stability and Trijunction Motion in Eutectic Solidification

We demonstrate by both experiments and phase-field simulations that lamellar eutectic growth can be stable for a wide range of spacings below the point of minimum undercooling at low velocity, contrary to what is predicted by existing stability analyses. This overstabilization can be explained by relaxing Cahn's assumption that lamellae grow locally normal to the eutectic interface.Comment: 4 pages, 5 eps figure