There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

The phase-field method has become an important and extremely versatile technique for simulating microstructure evolution at the mesoscale. Thanks to the diffuse-interface approach, it allows us to study the evolution of arbitrary complex grain morphologies without any presumption on their shape or mutual distribution. It is also straightforward to account for different thermodynamic driving forces for microstructure evolution, such as bulk and interfacial energy, elastic energy and electric or magnetic energy, and the effect of different transport processes, such as mass diffusion, heat conduction and convection. The purpose of the paper is to give an introduction to the phase-field modeling technique. The concept of diffuse interfaces, the phase-field variables, the thermodynamic driving force for microstructure evolution and the kinetic phase-field equations are introduced. Furthermore, common techniques for parameter determination and numerical solution of the equations are discussed. To show the variety in phase-field models, different model formulations are exploited, depending on which is most common or most illustrative.

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An introduction to phase-field modeling of microstructure evolution

The term superhydrophilicity is only 11–12 years old and was introduced just after the explosion of research on superhydrophobic surfaces, in response to the demand for surfaces and coatings with exceptionally strong affinity to water. The definition of superhydrophilic substrates has not been clarified yet, and unrestricted use of this term to hydrophilic surfaces has stirred controversy in the last few years in the surface chemistry community. In this review, we take a close look into major definitions of hydrophilic surfaces used in the past, before we review the physics behind the superhydrophilic phenomenon and make recommendation on defining superhydrophilic surfaces and coatings. We also review chemical and physical methods used in the fabrication of substrates on surfaces of which water spreads completely. Several applications of superhydrophilic surfaces, including examples from the authors' own research, conclude this review.

Hydrophilic and superhydrophilic surfaces and materials

This paper provides a review of superhydrophobicity and related phenomena (superoleophobicity, omniphobicity, self-cleaning) induced by surface micro- and nanostructuring. The classical approaches to superhydrophobicity using the Young, Wenzel, and Cassie–Baxter models for the contact angle (CA) are presented. After that, the issues that are beyond the Wenzel and Cassie–Baxter theories are discussed, such as multiscale effects, 1D vs. 2D interactions, the effects of contact line, size of roughness details, curvature, and CA hysteresis dependence on roughness. New potential applications of superhydrophobicity are reviewed, such as new ways of energy transition, antifouling, and environment-friendly manufacturing.

Superhydrophobic surfaces and emerging applications: Non-adhesion, energy, green engineering

Contact angle hysteresis is an important physical phenomenon. It is omnipresent in nature and also plays a crucial role in various industrial processes. Despite its relevance, there is a lack of consensus on how to incorporate a description of contact angle hysteresis into physical models. To clarify this, starting from the basic definition of contact angle hysteresis, we introduce the formalism and models for implementing contact angle hysteresis into relevant physical phenomena. Furthermore, we explain the influence of the contact angle hysteresis in physical phenomena relevant for industrial applications such as sliding drops, coffee stain phenomenon (in general evaporative self-assembly), and curtain and wire coating techniques

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Contact angle hysteresis: a review of fundamentals and applications

No-slip boundary condition in finite-size dissipative particle dynamics

We propose a novel approach to describe wetting of plane solid surfaces by liquid drops. A two-dimensional nonconserved phase field variable is employed to distinguish between wetted and nonwetted regions on the surface. The imbalance in the Young's force provides for the exchange of relative stability of the two phases. The three-phase contact line tension arises from the gradient energy and contact angle hysteresis from the kinetic coefficient. Using this theory, we discuss contact angle hysteresis on chemically heterogeneous surfaces. We show significant departure from the classical Cassie theory, which is attributed to defect pinning of the continuous triple line.

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Phase field modeling of hysteresis in sessile drops.

In a gas turbine including a nozzle retaining ring having a first annular axially sealing surface and a shroud segment having an axial registering second surface. To minimize or prevent leakage flow between the retaining ring and shroud segments, a generally U-shaped seal having reversely folded U-shaped marginal portions is received in a cavity formed in the second surface. At operating conditions, the marginal portions seal against the base of the cavity and the first surface of the retaining ring to prevent leakage flow past the retaining ring/shroud segment interface. To install the seal, the seal body is first compressed and maintained in a compressed state by applying one or more wraps about the seal body and an epoxy is used to secure the seal when compressed in the cavity. At operating temperatures, the retention means releases the seal to engage marginal portions against opposite sealing surfaces of the shroud segments and retaining ring.

Seal for gas turbine nozzle and shroud interface

In this paper we develop a free energy function for a Cu–Al–Ni alloy undergoing cubic–orthorhombic phase transitions. We use the irreducible Lagrangian strain polynomial invariants of the cubic austenite parent phase to construct a polynomial expansion for the Helmholtz free energy. Our expansion retains quartic terms of the strain components in order to describe the two-phase material—the cubic austenite phase and six variants of the orthorhombic martensite phase. The coefficients of the free energy polynomial function are temperature dependent and are fitted to appropriate experimental data for austenite and martensite phases from literature. The resulting Helmholtz free energy function is given by (3.20), (4.11). We examine the response predicted by the model for shear in the twinning directions.

Srikanth Vedantam

Papers

No-slip boundary condition in finite-size dissipative particle dynamics

Phase field modeling of hysteresis in sessile drops.

Seal for gas turbine nozzle and shroud interface

A Helmholtz free-energy function for a Cu–Al–Ni shape memory alloy

Phase field study of the effect of grain boundary energy anisotropy on grain growth