Phase Boundary Energy Research Articles

Modulated self-assembly of 4,4′-diphenyltetrathiafulvalene molecules on highly oriented pyrolytic graphite byn-tetradecane solvent

We report the formation of a binary-component self-assembled monolayer (SAM) comprising4,4′-diphenyltetrathiafulvalene(DP-TTF) and n-tetradecane (n-C14H30) molecules with periodic strip-like phase separation structures on a highly orientedpyrolytic graphite (HOPG) surface. Scanning tunneling microscopy (STM) imaging revealsthat ordered DP-TTF single- and double-lamella are periodically tuned by orderedn-C14H30 single- and double-lamella, respectively. This finding can be qualitatively understood interms of a phase field model, in which the interplay of three ingredients, including freeenergy of the binary-component solution monolayer, phase boundary energy andsurface stress, determines the final equilibrium sizes of the ordered DP-TTF andn-C14H30 phases in the binary-component SAM. Furthermore, anisotropy of thesurface stress breaks the symmetry of the substrate and causes then-C14H30 molecules to arrange along preferential substrate⟨010⟩ directions. Theorientation of the n-C14H30 molecule stripes further guides the directions of the DP-TTF lamellar structures.In addition, scanning tunneling spectra (STS) of the individual DP-TTF andn-C14H30 molecules in the ordered monolayer show a remarkable difference inI(V) curves on the HOPG substrate.

Stability of multi-component epilayers and nanopattern formation

A uniform multi-component epilayer may lose stability under the combined action of spinodal decomposition and epilayer–substrate interaction, separating into multiple phases. The phases may further self-organize into regular patterns. This paper investigates the compositional stability of a ternary epliayer and the subsequent emergence of nanoscale patterns. Multiple energetic forces and kinetic processes involving phase separation, phase coarsening and phase refining are incorporated into a continuous phase field model. Linear stability analysis is performed by perturbing a uniform concentration field into a sinusoidal field with small amplitude and arbitrary wavelength. The analysis shows that the epilayer–substrate interaction counteracts the coarsening effect of phase boundary energy and may lead to the formation of steady nanoscale patterns. Detailed analysis also reveals the interaction of multi-phases and its effect on the stability condition. Numerical simulation of evolving concentration field is discussed at the end of the paper. The simulations show that the pattern formation process of multi-component epilayers involves remarkably rich dynamics.

Simulation on nanoscale self-assembly of ternary-epilayers

This paper simulates the self-assembly of a ternary epilayer on an elastic substrate. The phases coarsen to reduce the phase boundary energy and refine to reduce the elastic energy. They may order into nanoscale patterns to minimize the total free energy. We combine spinodal decomposition and surface stress into a continuum phase field model. A set of nonlinear diffusion equations couples the two concentration fields in the epilayer and the stress field in the substrate. An efficient simulation technique based on the Fast Fourier Transform and the semi-implicit method is developed. Numerical simulations reveal remarkably rich dynamics in the self-assembly of ternary epilayers and suggest significant degree of experimental control in the growth of nanoscale superlattices.

Self-organized nanostructures in multi-phase epilayers

Recent experiments have shown that multi-phase epilayers may self-organize into orderednanoscale patterns on a solid substrate. In this paper, we present a phase field model forthe self-assembly of three-phase epilayers. Nanoscale patterns are developed by twocompeting actions: coarsening due to phase boundary energy and refining due tosubstrate-mediated elastic interaction. The continuum phase field approach leads to a set ofnonlinear diffusion equations, which couples the two concentration fields in the epilayer andthe stress field in the substrate. Numerical simulations reveal remarkably rich dynamics inthe self-assembly of multi-phase epilayers. An epilayer may evolve into various patterns,suggesting a significant degree of experimental control in growing nanoscale structures.

Dynamics of nanoscale self-assembly of ternary epilayers

This paper models the self-assembly dynamics of a ternary epilayer on an elastic substrate. The phases coarsen to reduce the phase boundary energy and refine to reduce the elastic energy. They may order into nanoscale patterns to minimize the total free energy. We combine the continuum phase field model of spinodal decomposition and the surface stress. A set of nonlinear diffusion equations couple the two concentration fields in the epilayer and the stress field in the substrate. Numerical simulations reveal remarkably rich dynamics and suggest the exciting possibility of controlling a nanoscale self-assembly process.

Patterning Nanoscale Structures by Surface Chemistry

This letter reports on a simulation of the nanoscale self-assembly process of a two-phase monolayer on an elastic substrate. Two competing actions determine the phase sizes and their spatial ordering. The phase boundary energy tends to coarsen the phases, while the concentration-dependent surface stress tends to refine the phases. A continuum phase field model is developed, which combines spinodal decomposition, surface stress and surface chemistry. The simulation shows that the self-assembly process can be guided by tuning the surface chemistry of a substrate. An epilayer may evolve into various nanoscale patterns in such a way.

Passing stiffness anisotropy in multilayers and its effects on nanoscale surface self-organization

A binary monolayer adsorbed on a solid surface can separate into distinct phases that further self-assemble into various two-dimensional patterns. The surface stresses in the two phases are different, causing an elastic field in the substrate. The self-organization minimizes the combined free energy of mixing, phase boundary, and elasticity. One can obtain diverse patterns by using substrates with various crystalline symmetries. Consider the pattern of a set of periodic stripes. The stripe orientation depends on the anisotropy in surface stress, substrate stiffness, and phase boundary energy. A more powerful and flexible way is to use a layered substrate. Surface properties designed for the applications of those patterns can be obtained by choosing appropriate materials and structures for the monolayer and the top layer of the substrate. The subsequent layers of the substrate provide the required stiffness anisotropy, the effect of which is passed to the monolayer patterns through the elastic field. We solve the elastic field in the anisotropic, heterogeneous, three-dimensional half-space by using the Eshelby–Stroh–Lekhnitskii formalism and the Fourier transformation. Depending on the thicknesses and the degrees of the stiffness anisotropy of the substrate layers, the lowest energy stripes can have tunable equilibrium size and orientation. We also discuss other possibilities of manipulating the phase patterns by engineering the elastic field.

The orientation of the self-assembled monolayer stripes on a crystalline substrate

Two-phase monolayers adsorbed on crystalline substrates can form many patterns. After reviewing the experimentally observed patterns on various substrates, we extend a thermodynamic theory to account for the anisotropy in surface stress, substrate stiffness, and phase boundary energy. We solve the elastic field in the anisotropic substrate by using the Stroh formalism. We then focus on the pattern of periodic stripes, and determine the orientation of the stripes that minimizes the free energy. As an example, we examine in detail the (1 1 0) surface of a cubic crystal. Depending on the parameters that characterize anisotropy, the stripes can orient along either [ 1 ̄ 1 0] , or [0 0 1] , or certain directions off the two crystalline axes. The transition between these orientations can be of either first or second order. The predications point to additional experiments that are needed to further the understanding.

Symmetry breaking in self-assembled monolayers on solid surfaces. II. Anisotropic substrate elasticity

On a solid surface a monolayer mixture may separate into two phases. The phases can self-assemble into a variety of patterns, with the feature size on the scale of 1‐100 nm. This paper studies how the elastic anisotropy of the substrate affects the patterns. The substrate is taken to be of cubic crystal symmetry, although the method developed can be readily extended to treat substrates of any other crystalline symmetry. The surface stress differs from one phase to the other, and the difference refines the phases to reduce the elastic energy. The phase boundary energy tends to coarsen the phases. The two competing effects set the phase sizes and the phase patterns. The anisotropy of the substrate elasticity breaks the symmetry of the system. We formulate a phase field model on the basis of this physical picture. The numerical simulation shows that stripes tend to orient along the two compliant directions of the substrate, forming a tweedlike pattern. A square lattice of dots can also be obtained when the anisotropy is weak and the average concentration sufficiently deviates from half filling. We interpret these findings in terms of free energy minimization.

Symmetry breaking in self-assembled monolayers on solid surfaces: Anisotropic surface stress

This paper models the self-assembly dynamics of a two-phase monolayer on an elastic substrate. The two phases coarsen to reduce the phase boundary energy and refine to reduce the elastic energy. To minimize the total free energy, the two phases can order into nanoscale patterns. We combine the continuum phase field model of spinodal decomposition and the anisotropic surface stress. The numerical simulation shows various patterns, such as interwoven stripes, parallel stripes, triangular lattice of dots, and herringbone structures. The surface stress anisotropy causes a transition from the parallel stripes to the herringbone structures. We show that this symmetry breaking mesophase transition obeys the classical theory of Landau.

Dynamics of nanoscale pattern formation of an epitaxial monolayer

A two-phase monolayer grown on an elastic substrate may form stripes or dots on the scale of nanometers. Sometimes these stripes and dots order into superlattices. This paper reports on a simulation on the basis of a model proposed by the authors recently. The size selection and spatial ordering result from two competing actions: the phase boundary energy tends to coarsen the phases, and the concentration-dependent surface stress tends to refine the phases. A nonlinear diffusion equation couples the concentration field in the epilayer and the stress field in the substrate. The simulation reveals remarkably rich dynamics. An epilayer may evolve into various patterns, suggesting a significant degree of experimental control in growing nanoscale superlattices, just as in growing atomic crystals.

Forces that Drive Nanoscale Self-assembly on Solid Surfaces

Experimental evidence has accumulated in the recent decade that nanoscale patterns can self-assemble on solid surfaces. A two-component monolayer grown on a solid surface may separate into distinct phases. Sometimes the phases select sizes about 10 nm, and order into an array of stripes or disks. This paper reviews a model that accounts for these behaviors. Attention is focused on thermodynamic forces that drive the self-assembly. A double-welled, composition-dependent free energy drives phase separation. The phase boundary energy drives phase coarsening. The concentration-dependent surface stress drives phase refining. It is the competition between the coarsening and the refining that leads to size selection and spatial ordering. These thermodynamic forces are embodied in a nonlinear diffusion equation. Numerical simulations reveal rich dynamics of the pattern formation process. It is relatively fast for the phases to separate and select a uniform size, but exceedingly slow to order over a long distance, unless the symmetry is suitably broken.

Phase boundary structure in Nd–Fe–B sintered magnets

The interface between tetragonal Nd2Fe14B phase and Nd-rich grain boundary phase in Nd–Fe–B sintered magnets was investigated with high resolution transmission electron microscope. Selected area diffraction patterns of the Nd-rich phase can be indexed by face-centered-cubic (fcc) structure with the lattice constant of a=0.548 nm. The orientation relationships between the two phases contacted directly with each other approximately agree with either of the following formulas: (110)tetra.//(110)fcc, [001]tetra.//[22̄3]fcc, or (110)tetra.//(010)fcc, [001]tetra.//[1̄02]fcc. These preferred orientations would be the results which minimize the phase boundary energy during postsintering heat treatment. In addition, it may change the local crystal field as well as the strain at the outermost Nd3+ ion site of the tetragonal grain both of which affect the coercivity of the magnet.

Mechanical Properties and Deformation Behavior of the Interfaces of Dissimilar Metals

Bimetallic joints with clean interfaces, obtained by a special cold welding method, are used as a model of phase boundaries for investigation their strength, deformation behavior, fracture mode and relaxation processes at the interface. Pairs of well-soluble and insoluble metals as well as pairs forming chemical compounds and eutectics were choosen for investigation. It has been shown that differences in the mechanical behavior of the interfaces are related to differences in phase boundary energy and other thermodynamic parameters as well as to mechanoactivation of both chemical interaction and structure formation.

Mechanical properties and accommodation processes on metallic interfaces

Bimetallic joints Al Pb , Al Sn , Pb Sn , Pb Pb , etc. with clean interfaces, obtained by a special cold welding method, are used as a model of phase boundaries for investigation of accommodation processes and strength properties of interfaces. To reduce the volume diffusioninduced relaxation processes, investigations were carried out at relatively low temperatures 0.1–0.5 T m. The role of surface diffusion and the effect of phase boundary energy on the healing of micropore ensembles on interfaces was investigated. The accommodation processes on the phase boundaries are considered as a result of mechanoactivation of both the interaction and structure formation of such non-equilibrium systems. It has been shown that the nanocrystalline state promotes the healing of the micropores at the interface.

The growth of holes in plates of cementite

Isotropic and anisotropic of circular and elliptic holes in plates of cementite of lamelar pearlite is discussed. The driving force for hole growth is obtained either from the difference in surface energy of the plates or from the inverses of the curvature of the holes. Two types of model are developed. In the first, constant thickness of the plate and, in the second, constant volume of the plate is assumed. For both models it is shown that the growth of holes is possible if the holes are of minimum size. The dependence of the size of this critical hole on the shape of the holes and on the ratio of half the distance of the holes on the plate to the thickness of the plate is discussed. Using the proposed models, experimental observations on the spheroidization of lamellar pearlite can be explained. A short discussion is given on the anisotropy of the ferrite-cementite phase boundary energy.

Integral treatment of subcooled forced convection film boiling on a flat plate

Subcooled forced convection film boiling on a flat plate has been analysed by means of an integral method. Following the two phase boundary layer theory, the momentum and energy equations for both liquid and vapor layers are considered along with the compatibility conditions on the liquid-vapor interface. Subsequently, the governing equations are reduced to a set of algebraic equations which can readily be solved for given parameters. Comparison of the present solution with the Cess and Sparrow solution reveals an excellent performance of the present solution procedure. The effects of superheating, subcooling and liquid Prandtl number on the hydrodynamic and heat transfer characteristics are fully discussed. Furthermore, the asymptotic formulas are derived for the local Nusselt number and skin friction coefficient through a careful examination of the physical limiting conditions.

The crystallography of phase interfaces in the NiCr alloy system

We have determined, by transmission electron microscopy, the crystallographic relationships between the Ni-rich and Cr-rich solid solutions composing the preferred interphase interfaces in the Ni-Cr alloy system. When the interface is the result of the eutectic reaction or solid-state matrix nucleation, good atomic matching between phases governs the development of the preferred interface plane. The crystallographic relationships observed for these cases are: (1 1 ̄ 1 ̄ ) Ni(γ 1) ∥(01 1 ̄ ) Cr(α 1) ∥ eutectic interface plane; [211] Ni( γ 1) ∥[311] Cr( α 1)∥ growth direction; and (1 1 ̄ 1 ̄ ) Ni(γ 2)∥(01 1 ̄ ) Cr(α 1) ∥ Widmanstätten precipitate interface; [101] Ni( γ 2) ∥[100] Cr( α 1). When precipitate nucleation is defect-aided, as in the case of the cellular precipitation of the Cr-rich phase, low atomic mismatch combined with the high probability of nucleation on {311} Ni favor the following relationships: (dash11̄3) Ni(gg) ∥(12̄1̄) Cr(α) ∥cellular precipitate interface; [211] Ni(γ) ∥[311]Cr(γ). The eutectic interphase boundaries are composed of three dislocation sets; the dislocation spacings are characteristic of a semi-coherent interface. The phase boundary energy, γ 1 α 1, computed from solidification data and dislocation spacing is about 300 ergs/cm 2. The precipitate-matrix interfaces are probably semi-coherent also.

A study of optimal phase boundaries: the case of exsolved alkali feldspars

A general theory of grain and phase boundaries (the O-lattice theory) is further developed and tested on alkali feldspars with exsolution lamellae, since measurements exist for the structures of the two-phase system (perthitic feldspars) as well as for the orientation of the phase boundary. It is shown that in this case the adaptation of two monoclinic structures is energetically preferable to the adaptation of a monoclinic and a triclinic one. The phase boundary energy is markedly lower in the former case. Thus, a pseudo-monoclinic structure is produced out of the triclinic by periodic submicroscopic twinning. The calculated orientation of the phase boundary is in close agreement with the measurements.