Motions of Microscopic Surfaces in Materials
Introduction
Soap bubbles show up in kitchens, science museums, and popular books (e.g., Isenberg, 1978). There has long been a tradition of drawing analogies between soap films and microscopic surfaces in solids. The analogy, however, can be misleading. The air pressure in each bubble is uniform and relates to the bubble volume. The shapes of an assemblage of bubbles minimize the total film area for the given volume of every bubble. The shapes change when air is blown into the bubbles or diffuses across the films.
In solids, there exist phase boundaries, grain boundaries, domain walls, and bi-material interfaces. The stress in each solid grain is usually non-uniform, and the total surface area need not be minimal for given grain volumes. In addition to surface tension, the free energy results from stress, electric field, and composition gradient, etc. Kinetic processes include diffusion, creep, and reaction.
The motion of the microscopic surfaces affects material processing and performance. For a bulk material, an overall knowledge of the structure, such as the grain size distribution and pore volume fraction, is often adequate. For a film or a line, where the grain size is comparable to the film thickness and linewidth, an overall knowledge of structure is inadequate; for example, in submicron aluminum interconnects, the electromigration damage relates to structural details, e.g., crystalline texture, individual grain-boundary orientation (Thompson and Lloyd, 1993). In cases like this, the internal surfaces are better viewed as components of one single structure.
We can now analyze deformation in complex structures using general-purpose computer codes. It would help many technical innovations if we could do the same for the evolving structures in materials. With this in mind, this article reviews the recent development of an approach that treats surface motion in a way that resembles the finite element analysis of deformation. Attention is focused on two mass transport mechanisms: migration of, and diffusion on, an interface. Examples are also given for other mass transport mechanisms.
At the heart of the approach is a weak statement that combines the kinetic laws and the free energy variation associated with virtual surface motion. On one hand, this weak statement reproduces the differential equations of Herring (1951) and Mullins (1957). On the other hand, this weak statement forms the basis for various Galerkin-type methods. In the latter, a surface is described with a finite number of generalized coordinates, and the Galerkin procedure reduces the weak statement to a set of ordinary differential equations that evolve into the generalized coordinates.
Depending on one’s purpose, one may describe a surface with either a few or many degrees of freedom. To study certain global aspects of the surface motion, one may describe the surface with a few degrees of freedom. Ideas in low-dimensional nonlinear dynamics apply. Even with a linear kinetic law, surface evolution is highly nonlinear because of large shape and topology changes. The surface may undergo instabilities and bifurcations.
Rigorously, a surface has infinitely many degrees of freedom. To resolve local details, one must describe the surface with many degrees of freedom. A systematic approach is to divide the surface into many small, but finite, elements and follow the motion of the nodes of the elements. The Galerkin procedure gives a viscosity matrix that connects the generalized forces and the generalized velocities. The procedure is analogous to the finite element analysis of deformation.
Most sections of this article may be read independently. The main exceptions are II Interface Migration: Formulation, V Diffusion on Interface: Formulation, which formulate, respectively, interface migration and interface diffusion. The subjects of all sections should be clear from the Table of Contents. Free energy is used throughout the article to study isothermal processes. (An entropy-based formulation is necessary if heat transfer plays a part.) The treatment is phenomenological with few references to the underlying atomic processes. Such continuum models are indispensable because a microstructural feature often contains a huge number of atoms. Technical processes are used to motivate the discussion, but the emphasis is on basic principles and simple demonstrations. Analytical solutions of several idealized models are included; they shed light on more complex phenomena, and may also serve as benchmark problems for general-purpose codes in the future. No attempt, however, has been made to review the literature exhaustively. By focusing on the principles and demonstrations, the reader should grasp what this line of thinking has to offer, and integrate it to his or her own way of thinking.
Section snippets
Interface Migration: Formulation
This section demonstrates the basic principles by examining a classical model with very few ingredients. An interface separates either two materials, or two phases of the same atomic composition, or two grains of the same crystalline structure. The free energy that drives the interface migration has contributions from many origins. This section includes only the interface tension, and the free-energy difference between the two phases in bulk.
Many kinetic processes may determine the velocity of
Interface Migration Driven by Surface Tension and Phase Difference
This section gives examples of interface migration under surface tension and free energy density differences between the two phases. Finite element schemes have been formulated on the basis of the weak statement (Cocks and Gill, 1995; Du et al., 1996; Sun et al., 1997). It is too early to judge them critically. Instead, this section gives an elementary demonstration of the Galerkin procedure, and describes several analytical solutions.
Interface Migration in the Presence of Stress and Electric Fields
In many material processes, elastic and electrostatic fields allow additional means of free-energy variation. For example, during a phase transition, the difference in the crystalline structures of the two phases induces a stress field (e.g., Eshelby, 1970, Abeyaratne and Knowles, 1990, Lusk, 1994, Rosakis and Tsai, 1994). In a polycrystalline film, grains of different orientations have different elastic energy densities due to elastic or plastic anisotropy (e.g., Sanchez and Arzt, 1992, Floro
Diffusion on Interface: Formulation
This section formulates mass diffusion on an interface. The interface may be either a free surface, or a grain boundary. The diffusion species are taken to be electrically neutral, so that only mass conservation need be enforced. The free energy has the same contributions as before, e.g., surface tension, external work, and elastic energy.
Shape Change due to Surface Diffusion under Surface Tension
This section gives examples of shape changes motivated by surface tension. Most examples invoke surface diffusion as the only mass-transport mechanism. One example involves simultaneous grain-boundary migration and surface diffusion.
Diffusion on an Interface between Two Materials
An interface between two materials is a rapid diffusion path for impurity atoms and atoms of the two materials. This section concerns with the latter. Both materials are taken to be rigid. (Sofronis and McMeeking, 1994 considered the combined interface diffusion and matrix creep in composite materials, which will not be considered here.) On an Al-Al2O3 interface, for example, one expects that aluminum diffuses much faster than oxygen, the latter being tied by the stronger atomic bonds. The
Surface Diffusion Driven by Surface- and Elastic-Energy Variation
A small crystal can sustain a high stress without fracture or plastic deformation. At an elevated temperature, the elastic energy can motivate mass diffusion. For example, when a film is deposited on a substrate with similar crystal structure having a few percent difference in lattice constant, the film strains to match the substrate lattice constant. The stress in the film would exceed 1 GPa were all relaxation processes suppressed. When the film is thick, the stress is relieved by dislocations
Electromigration on Surface
Interconnects in integrated circuits are made of aluminum alloys. They have small cross sections (less than 1 μm wide and about 0.5 μm thick), carry electric current up to 1010 A/m2, and operate near half of aluminum’s melting temperature (933° K). The flowing electrons exert a force on aluminum atoms (i.e., the electron wind force), motivating aluminum atoms to diffuse. The phenomenon—mass diffusion directed by electric current, known as electromigration—causes reliability problems in integrated
Acknowledgments
The writer is grateful to the National Science Foundation for a Young Investigator Award, to the Humboldt Foundation and the Max Planck Society for financing a sabbatical leave at the Max Planck Institute in Stuttgart hosted by Directors M. Ruhle and E. Arzt, and to Advanced Micro Devices for a grant under the supervision of Dr. J. E. Sanchez. Part of the work reviewed here was supported by ARPA through a URI contract N-0014-92-J-1808, by ONR through contract N00014-93-1-0110, and by NSF
References (113)
- et al.
On the driving traction acting on a surface of strain discontinuity in a continuum
J. Mech. Phys. Solids
(1990) - et al.
Void growth and collapse in viscous solids
- et al.
Interplay of sintering microstructures, driving forces, and mass transport mechanisms
J. Am. Ceram. Soc
(1989) - et al.
Stress singularities along a cycloid rough surface
Int. J. Solids Struct
(1993) The structure of constitutive laws for the sintering of fine grained materials
Acta Metall. Mater
(1994)The determination of the elastic field of an ellipsoidal inclusion, and related problems
Proc. R. Soc. London
(1957)- et al.
Instability of a biaxially stressed thin film on a substrate due to material diffusion over its free surface
J. Mech. Phys. Solids
(1993) - et al.
Simulation of thin film grain structures—II. Abnormal grain growth
Acta Metall. Mater
(1992) A boundary perturbation analysis for elastic inclusions and interfaces
Int. J. Solids Struct
(1991)Stress analysis of holes in anisotropic elastic solids: Conformal mapping and boundary perturbation
Q. J. Mech. Appl. Math
(1992)