Convexity properties of the surface tension and equilibrium crystals

We study the thermodynamic limit of the orientation-dependent surface tension. Under general conditions, which we show to hold true for a large class of lattice systems, we prove that the limit exists and that it satisfies some convexity properties related to the pyramidal inequality introduced by R. L. Dobrushin and S. B. Shlosman⁽¹⁾. We discuss some consequences of these results for the equilibrium crystal shape.     

Surface tension,percolation, and roughening

We describe inequalities relating to the interface between coexisting phases of Ising ferromagnets. Some implications for the nature of the roughening transition are discussed.Supported by National Science Foundation Grant No. MCS78-01885 at Princeton University and No. PHYS78-15920 at Rutgers University.Part of this work was done while Jean Bricmont was at the Mathematics Department of Princeton University and Joel L. Lebowitz at the Institute for Advanced Study, Princeton.     

Asymptotic degeneracy of the transfer matrix spectrum for systems with interfaces: Relation to surface stiffness and step free energy

Two- and three-dimensional Ising-type systems are considered in the finite-cross-section cylindrical geometry. An interface is forced along the cylinder (strip in 2d) by the antiperiodic or +– boundary conditions. Detailed predictions are presented for the largest asymptotically degenerate set of the transfer matrix eigenvalues. For rough interfaces, i.e., for 0<T<T _c in 2d,T _R<T<T _c in 3d, the eigenvalues are split algebraically, and the spectral gaps are governed by thesurface stiffness coefficient. For rigid interfaces, i.e., 0<T<T _R in 3d, the eigenvalues are split exponentially, with the gaps determined by thestep free energy.     

Surface free energy of polypropylene and polycarbonate solidifying at different solid surfaces

Advancing and receding contact angles of water, formamide, glycerol and diiodomethane were measured on polypropylene (PP) and polycarbonate (PC) sample surfaces which solidified at Teflon, glass or stainless steel as matrix surfaces. Then from the contact angle hystereses (CAH) the apparent free energies of the surfaces were evaluated. The original PP surface is practically nonpolar, possessing small electron donor interaction (), as determined from the advancing contact angles of these liquids. It may result from impurities of the polymerization process. However, it increases up to 8-10 mJ/m² for PP surfaces contacted with the solids. The PC surfaces both original and modified show practically the same . No electron acceptor interaction is found on the surfaces.The of modified PP and PC surfaces depend on the kind of probe liquid and contacted solid surface. The modified PP values determined from CAH of polar liquids are greater than that of original surface and they increase in the sequence: Teflon, glass, stainless steel surface, at which they solidified. No clear dependence is observed between and dielectric constant or dipole moment of the polar probe liquids. The changes in of the polymer surfaces are due to the polymer nature and changes in its surface structure caused by the structure and force field of the contacting solid. It has been confirmed by AFM images.     

Study of surface cell Madelung constant and surface free energy of nanosized crystal grain

Surface cell Madelung constant is firstly defined for calculating the surface free energy of nanosized crystal grains, which explains the physical performance of small crystals and may be greatly beneficial to the analysis of surface states and the study of the dynamics of crystal nucleation and growth. A new approximative expression of the surface energy and relevant thermodynamic data are used in this calculation. New formula and computing method for calculating the Madelung constant α of any complex crystals are proposed, and the surface free energies and surface electrostatic energies of nanosized crystal grains and the Madelung constant of some complex crystals are theoretically calculated in this paper. The surface free energy of nanosized-crystal-grain TiO₂ and the surface electrostatic energy(absolute value) of nanosized-crystal-grain α -Al₂O₃ are found to be the biggest among all the crystal grains including those of other species.     

双轴向列相液晶的表面能

利用双轴向列相液晶表面能对指向矢和序参量矩阵的旋转不变性,构造了与指向矢^a,^b,^c对易取向轴^a0,^b0,^c0的偏离有关的表面能Fs(a,b,c),以及与序参量矩阵有关的表面能Fs(Q),并得到两种表达式的等价形式.在Fs(a,b,c)和Fs(Q)中分别有四个和两个独立参量.为讨论上述独立参量的物理意义,假设一液晶分子对表面相互作用的简单模型,将液晶分子沿易取向轴^b0,^c0的双轴性取向看成是受到沿^b0,^c0两相互垂直的平均场作用的结果.为测量表面锚泊系数,考虑了三种简单形变位型,即在外磁 关键词： 液晶 弹性能 表面能     

Eight-vertex model: Anisotropic interfacial tension and equilibrium crystal shape

The anisotropic interfacial tension of the eight-vertex model is found by a new method, which introduces two inhomogeneous systems. As the width of the system becomes large, a doublet of the largest eigenvalues of the row-row transfer matrix is asymptotically degenerate. The anisotropic interfacial tension is calculated from their finite-size correction terms in this limit. By the use of the anisotropic interfacial tension, the equilibrium crystal shape of the eight-vertex model is derived via Wulff's construction. The equilibrium crystal shape is represented as a simple algebraic curve. We discuss the close relation between the algebraic curve and the form of an elliptic function appearing in the expression of the interfacial tension.     

Simple inequality for the free energy of disordered systems

It is proved that the free energy of a disordered system described by a quadratic form in Bose or Fermi operators with random coefficients, calculated in the simplest approximation for the associated eigenvalue problem, gives the upper (Bose case) and lower (Fermi case) bounds for the exact free energy.     

On the interface at the equilibrium crystallization

Properties of an interface, created at some experimental conditions in the course of the equilibrium crystallization process, are theoretically investigated. Influence of quantum and thermal fluctuations on smoothing-roughening phase transitions is considered. The phase diagram illustrating these properties is found.     

Surface free energy of a finite system without interfaces at low temperatures

We derive the specific surface free energy for a rather general system at low temperatures that can be rewritten as a gas of non-interacting contours (polymers). To this end, we use a standard cluster expansion series for the system?s partition function. A specific regime of ‘weak’ boundary conditions is assumed to ensure that no interfaces or large droplets occur in the system. We illustrate the general results, using a simple lattice–gas model.     

Surface free energy of non-stick coatings deposited using closed field unbalanced magnetron sputter ion plating

Semiconductor IC packaging molding dies require wear resistance, corrosion resistance and non-sticking (with a low surface free energy). The molding releasing capability and performance are directly associated with the surface free energy between the coating and product material. The serious sticking problem reduces productivity and reliability. Depositing TiN, TiMoS, ZrN, CrC, CrN, NiCr, NiCrN, CrTiAlN and CrNiTiAlN coatings using closed field unbalanced magnetron sputter ion plating, and characterizing their surface free energy are the main object in developing a non-stick coating system for semiconductor IC molding tools. The contact angle of water, diiodomethane and ethylene glycol on the coated surfaces were measured at temperature in 20 °C using a Dataphysics OCA-20 contact angle analyzer. The surface free energy of the coatings and their components (dispersion and polar) were calculated using the Owens-Wendt geometric mean approach. The surface roughness was investigated by atomic force microscopy (AFM). The adhesion force of these coatings was measured using direct tensile pull-off test apparatus. The experimental results showed that NiCrN, CrN and NiCrTiAlN coatings outperformed TiN, ZrN, NiCr, CiTiAlN, CrC and TiMoS coatings in terms of non-sticking, and thus have the potential as working layers for injection molding industrial equipment, especially in semiconductor IC packaging molding applications.     

Surface Morphologies: Transient and Equilibrium Shapes

Many different well defined surface morphologies of crystalline materials have been studied in the past 50 years. One of the important and most interesting objectives has been to elucidate the physical origin of the anisotropic surface free energy and, in more recent time, the energetics of surface defects, such as monatomic steps and kinks, which are thermally generated at steps. Changes in surface morphology related to phase transitions, such as faceting or surface roughening, have also attracted much attention. Kinetic experiments dealing with non-equilibrium structures and transient shapes, have lead to quantitative transport properties, for example, surface diffusion coefficients. Fully equilibrated structures, such as 2D and 3D equilibrium crystal shapes, have been imaged to extract the orientation dependent step and surface free energies, even at various temperatures. New imaging techniques with atomic resolution (notably scanning tunneling microscopy) applied to transient as well as equilibrium structures have provided more data of increased accuracy in kinetics and defect/surface energetics. The present paper is an attempt of a brief review of this rapidly advancing field, emphasizing more recent results.     

能量公设与自由能判据的普遍化形式

从能量公设的角度论述了自由能及其判据的普遍化形式,并对其本质内涵进行了讨论．讨论认为：热力学系统中目由能的一般形式是与N种功类型相关的各种能形式之和,对孤立物体系在温度不变的情形下,不可逆过程总是导致总自由能的减少．通过实例说明了普遍化形式自由能判据的应用．     

Surface tension and universality in the three-dimensional Ising model

The amplitude ₀ of the interfacial free energy per unit area (or surface tension) of the body-centered-cubic Ising model is found using a direct monte carlo simulation technique. The combination ²/k_BT_c, where is the correlation length, is shown to agree within the precision of the simulations with a previously reported estimate for the simple cubic lattice. Evidence is also presented for the universality of the finite-size scaling amplitude for the surface tension.     

Effect of the surface free energy on the behaviour of surface and guided waves

The aim of this paper is to evaluate the influence of the surface free energy upon the propagation of the eigenmodes of structures, by studying successively (a) the Rayleigh wave for an elastic half-space, (b) the Lamb waves for an elastic layer, and (c) the guided modes for a tri-layer structure (e.g., metal/adhesive/metal). The surface free energy is a parameter which appears in the jump conditions of stresses and displacements at each interface, and which consequently modifies the eigenmodes, solutions of the boundary conditions system. As expected, the Rayleigh wave is dispersive and its velocity increases when the surface free energy increases. In the same way, the velocity of Lamb waves also increases except at normal angle of propagation where the surface free energy does not arise. Moreover, near the Rayleigh angle, the behaviour of the A₀ and S₀ Lamb modes varies strongly according to the surface free energy. Similar results are observed for the tri-layer structure.     

Surface order large deviations of phase interfaces for the continuum Widom-Rowlinson model in high density limit

We establish a surface order large deviation principle characterising, in the phase coexistence region, the exponential decay rates for the probabilities of macroscopic fluctuations of phase-separating interfaces for the continuum Widom-Rowlinson binary gas, with the thermodynamic and high fugacity limits taken simultaneously. The large deviation rate function is given by an isotropic surface energy functional and hence it attains its minimum for balls which are the most favourable shapes of ‘droplets’ of dominated phase within the ‘ocean’ of dominating phase.     

Vague concept of “reversible cleavage” in the theory of the surface tension of solids

Numerous derivations of the well-known Shuttleworth equation have been based on the unclear concept of “reversible cleavage” leading to the decisive step in any derivation - equalization of the surface free energy and surface stress. This is the key concept in contemporary surface thermodynamics of solids. But “cleavage” is not a surface process and, in this field, it cannot be a reversible operation. Besides, the “reversible cleavage” has no formal definition in the domain of the surface tension of solids that is an abnormal for any exact science. Consequently, this concept and all its corollaries including the Shuttleworth and generalized Lippmann equations have to be recognized as incorrect.     

Surface free energy of CrNx films deposited using closed field unbalanced magnetron sputtering

CrN_x thin films have attracted much attention for semiconductor IC packaging molding dies and forming tools due to their excellent hardness, thermal stability and non-sticking properties (low surface free energy). However, few data has been published on the surface free energy (SFE) of CrN_x films at temperatures in the range 20-170 °C. In this study CrN_x thin films with CrN, Cr(N), Cr₂N (and mixture of these phases) were prepared using closed field unbalanced magnetron sputtering at a wide range of Cr⁺² emission intensity. The contact angles of water, di-iodomethane and ethylene glycol on the coated surfaces were measured at temperatures in the range 20-170 °C using a Dataphysics OCA-20 contact angle analyzer. The surface free energy of the CrN_x films and their components (e.g., dispersion, polar) were calculated using the Owens-Wendt geometric mean approach. The influences of CrN_x film surface roughness and microstructure on the surface free energy were investigated by atomic force microscopy (AFM) and X-ray diffraction (XRD), respectively. The experimental results showed that the lowest total SFE was obtained corresponding to CrN at temperature in 20 °C. This is lower than that of Cr(N), Cr₂N (and mixture of these phases). The total SFE, dispersive SFE and polar SFE of CrN_x films decreased with increasing surface temperature. The film roughness has an obvious effect on the SFE and there is tendency for the SFE to increase with increasing film surface roughness.     

Effect of different solid matrixes on surface free energy of EGDMA and TRIM polymers

Advancing and receding contact angles of water, formamide, glycerol and diiodometane were measured on the two polymers; EGDMA (dimethacrylate of ethylene glycol) and TRIM (trimethacrylate-1,1,1-trihydroksymethylopropane) which were polymerized next to glass, silanized glass, stainless steel, mica and silicon surfaces as the matrices. Then from the contact angle hystereses (CAH) and van Oss, Good, Chaudhury (LWAB) approaches the apparent surface free energies were evaluated. The measured contact angles not only depend solely on the polymer chemical structure but also, to some extent, on the solid matrix next to whose surface the sample has polymerized. Surface free energy of the polymer samples calculated from the LWAB approach shows that they interact mainly by dispersive forces. The apparent surface free energy of the polymers calculated from the diiodomethane contact angles hysteresis is practically the same irrespective of the kind of the matrix used. Therefore it can be concluded that the observed weak polar interactions in the surface free energy of the samples depend on the polymer surface preparation. The AFM images show that the obtained surfaces are of different roughness. The RMS values of roughness range between 3.7-90.2 nm for EDGMA, and 5.3-124.5 nm for TRIM. However, as reported in literature, rather protrusions bigger than 1 μm may significantly affect the contact angles, especially the receding ones.