Interface between Superfluid and Solid He 4

Jeong

et al. 2003

Journal of Elec Materi

The transition in morphology of Ni 3 Sn 4 grains that formed at the interface between liquid Sn-3.5Ag (numbers are in wt.% unless specified otherwise) solder and Ni substrate has been observed at 250-650°C. The morphological transition of Ni 3 Sn 4 is due to the decrease of entropy of formation of the Ni 3 Sn 4 phase and has been explained well by the change of Jackson's parameter with temperature. According to the variation of solder joint strength with temperature, it decreased rapidly between 350°C and 450°C, where the thickness of the Ni 3 Sn 4 intermetallic compound (IMC) layer was around 6.5 µm. However, the solder joint strength decreased slowly with an increase of soldering time without a significant drop, although the thickness of the IMC was larger than 6.5 µm. The notable drop of solder joint strength and the fracture mode transition with increase of soldering temperature appears to come from excessive lateral growth of IMC grains between 350°C and 450°C.

Section: Discussionmentioning

confidence: 92%

Morphological transition of interfacial Ni3Sn4 grains at the Sn-3.5Ag/Ni joint

Jeong

et al. 2003

Journal of Elec Materi

American Journal of Physics

“…The angles are the most natural variables for a sphere, but we must be mindful of the periodic nature of φ ∈ (0, 2π] and the co-ordinate singularities at the poles θ = 0, π. An example is the shape of crystals in equilibrium with its liquid (for example, 4 He crystals in coexistence with the superfluid [15]) or vapor (for example, gold crystals [16]). Typical crystal shapes are not spherical and can be described by a non-trivial function R(θ, φ), which specifies the distance from the center of mass to a point on the crystal surface labeled by (θ, φ).…”

Section: Discussionmentioning

confidence: 99%

Making sense of the Legendre transform

Zia

Redish

McKay

2009

150

101

The Legendre transform is an important tool in theoretical physics, playing a critical role in classical mechanics, statistical mechanics, and thermodynamics. Yet, in typical undergraduate or graduate courses, the power of motivation and elegance of the method are often missing, unlike the treatments frequently enjoyed by Fourier transforms. We review and modify the presentation of Legendre transforms in a way that explicates the formal mathematics, resulting in manifestly symmetric equations, thereby clarifying the structure of the transform algebraically and geometrically. Then we bring in the physics to motivate the transform as a way of choosing independent variables that are more easily controlled. We demonstrate how the Legendre transform arises naturally from statistical mechanics and show how the use of dimensionless thermodynamic potentials leads to more natural and symmetric relations.

Proc. Natl. Acad. Sci. U.S.A.

“…Soon after that, c facets [the (0001) orientation in the hexagonal crystal lattice] were observed on 4 He crystals at slightly above 1 K by Landau et al (8). Afterward, Fisher and Weeks (9) argued that the equilibrium shape of all crystals (classical or quantum) is faceted at sufficiently low temperatures (see also ref.…”

mentioning

confidence: 98%

Faceting on ³ He crystals

Alles¹,

Бабкин²,

Jochemsen³

et al. 2002

It has been predicted by Landau that, ideally at low temperatures, crystals should show many different types of facets, i.e., flat smooth faces on their surface, but this so-called ''devil's staircase'' phenomenon has been difficult to observe experimentally. In this paper we describe our recent experiments, in which altogether 11 different types of facets have been identified on growing 3 He crystals at the temperature of 0.55 mK by using a unique lowtemperature Fabry-Pé rot interferometer. Previously only 3 types of facets had been seen in this system. We have also measured the growth velocities of different facets, and our interpretation of the obtained results yields the conclusion that 3 He has much stronger coupling of the liquid-solid interface to the crystal lattice than has been expected. After an introduction we present a short theoretical background about the equilibrium crystal shape and the roughening transitions, which is followed by the description of our experimental results and discussion.T he crystals that we find in nature have usually a polyhedral shape. Their surface is covered with smooth faces or facets that correspond to high-symmetry crystallographic orientations (see Fig. 1). These facets have developed on the crystal surfaces during growth when the crystals were formed. It must be pointed out that it is difficult to study the growth dynamics as well as the equilibrium shape of ordinary crystals. The difficulties are caused by big differences in entropy and density between the solid and the adjacent melt or vapor phases and finite thermal conductivities of these bulk phases. As a result, the relaxation processes are highly dissipative and the corresponding time scales extremely long. The equilibrium shapes have therefore been obtained with microscopic metallic crystals which after annealing of several days have showed facets surrounded by rough (rounded) areas on their surface (1).It was predicted by Landau in the middle of the last century that, in the limit of low temperatures, any ideal crystal should be completely faceted, with an infinite number of facets on its surface (2). This is the so-called ''devil's staircase '' phenomenon (3, 4). The most complicated crystals, however, have shown at most six types of facets. Only very recently Pieranski and his coworkers (5) were able to grow lyotropic monocrystals that revealed more than 60 types of facets on their surface, confirming experimentally the devil's staircase phenomenon. They interpreted their results as the coincidence of a quite large surface tension and interplanar distance with an exceptionally low elastic modulus. But with liquid lyotropic crystals, faceting can be studied in a very limited temperature range and it is hard to measure their equilibrium shape.The experimental situation is more promising in helium crystals, which exist at low temperatures and high pressures. The dynamics of helium crystals, which are surrounded by a superfluid with high thermal conductivity, can be extremely fast. For instance, 4 He crystals...