engineering of microstructures - scielo · engineering of microstructures ... the useful object...

Engineering of Microstructures

R.T. DeHoff

Department of Materials Science and Engineering,University of Florida, Gainesville, Florida

Received: August 15, 1998; Revised: March 30, 1999

Structure is at the heart of the materials science paradigm connecting processing with properties.In the hierarchy of structures that exist in materials microstructure offers the richest variety ofstructural arrangements. This variety is often conveniently accessible, e.g., simply by heat treatmentor mechanical deformation. Exploration of the relation between properties and microstructure servesto establish a target range of microstructural states that will perform. In order to attain a targetmicrostructure it is necessary to understand what microstructures are, and how they evolve inprocessing. This presentation focuses upon the set of tools that must be combined to achieve thiscontrol:

1. Geometry2 Thermodynamics3. Kinematics4. Kinetics.The content of these tools is reviewed briefly and their uses illustrated in developing an

understanding of how microstructures evolve. In this development an attempt is made to carry thedescription of each microstructural process as far as possible without making simplifying assump-tions. The study of microstructures with this rigorous point of view was termed by F.N. Rhines,“microstructology”.

Keywords: microstructure

1. Introduction

Geometry. The spatial distribution of things. The heartof the materials science paradigm, Fig. 1. Processing deter-mines structure; structure determines properties. Structureis the spatial distribution of things seen and unseen.

The scientific content of materials science is based upona hierarchy of structures:

Nuclear structure determines nuclear properties.Atomic structure consists of a spatial distribution of

subatomic particles.

Molecular structure extends this spatial distribution togroups of atoms and dominates the chemical behavior ofthe molecules.

Crystal structure for metals, alloys, ceramics and poly-mers is the spatial distribution of atoms on three dimen-sional lattices.

Electronic structure is visualized as the spatial distribu-tion of the probability density of an electron cloud, isconceived for atoms, molecules and crystals.

Defect structure in crystals includes point, line andsurface defects.

Microstructure influences virtually all aspects of thebehavior of materials.

Macrostructure is the spatial distribution of material inthe useful object that is the goal of the endeavor.

Materials science first adapted from physics or devisedthe conceptual content of these structures, then discoveredways to obtain information about that content, and is nowlearning to use that information to predict the response ofmaterials to processing, and the properties that the proc-essed material exhibits.

Review Article

Materials Research, Vol. 2, No. 3, 111-126, 1999. © 1999

Figure 1. The matrials science paradigm shows that structure connectsprocessing with properties.

For a given piece of material microstructure is the mostrichly variable, and thus the most susceptible to control, ofthe elements in the materials structural hierarchy. Further,access to this richness is frequently conveniently availablethrough very simple processes. For example a completerevision of the microstructure of a material may beachieved by simply changing its temperature. It is this richvariability, the attendant potential to produce a particularmicrostructural state with desired properties, and the easewith which this control can frequently be achieved, thatputs microstructure at the focal point of materials science.

This paper deals with half of the materials scienceparadigm: the selection of the material and processingcombination with the objective of engineering a targetmicrostructural state. The other half of the paradigm, theconnection of microstructure to properties and perform-ance so that the nature of the target microstructural statemay be identified, is a subject of equal importance, sophis-tication and complexity. However, this important subjectis outside the scope of this presentation.

A breadth of experience with particular materials andthe accumulating information and understanding about theoperation of the components in the hierarchy of structureshas brought the field to the point where microstructures canbe produced by design. Once the realm of the target micro-structure has been identified, the key to the design ofmicrostructures is an understanding of microstructural evo-lution. This requires an understanding of the processes thatare involved in bringing about a change in the spatialdistribution of the geometric elements that make up amicrostructure. This paper explores the tools that are re-quired to understand how microstructures evolve. Rhineshas termed the development of these tools and their knowl-edgeable application to understand how microstructuresarrive at a given state microstructology1. He believed thatthose who study microstructures should strive to be quan-titative in this endeavor and to take the application of thesetools as far as is possible before introducing simplifyingassumptions. These tools are:

1. Geometric concepts that describe the microstructuralstate, quantified by the application of stereology;

2. Thermodynamics and its implementation in phasediagrams which provide the context within which proc-esses occur;

3. Kinematics of evolving microstructures that relatelocal boundary motions to global geometric change; and

4. Kinetics of the processes that operate in the kinemat-ics.

The role of each of these tools in the design process willbe examined. Of course the tool that finds the most use inmicrostructural design is the microscope, whether it beTEM, SEM, the light microscope, or other versions ofinstruments that produce images of microstructures.

The first decision that will be made in the design processis a specification of the chemistry of the material. Thisprimal decision has the widest range of variables of all ofthe choices that are made in microstructural design. Howmany elements, which elements, and how much of each hasa profound effect on all subsequent possibilities. All of thecomponents in the materials structural hierarchy are cir-cumscribed first by this list of elements, and their relativeamounts, from which the material is made. Ultimately,these elements are gleaned from the earth’s crust. The verysophisticated and heroic processes that convert ores intomaterials are beyond the scope of this presentation. Ourinterest is focused upon structural evolution in solidifica-tion and in the solid state that follows.

The “microstructure” begins with the first nuclei of thesolid phase that forms in the liquid melt. The subsequentpath of microstructural evolution for the material is highlyvariable depending on the sequence of thermal, mechani-cal, chemical and physical environments in which the ma-terial is placed. The microstructural state at each pointalong the path is the precursor to each subsequent state. Andso the system evolves.

This development begins with a discussion of the con-cept of the microstructural state of a given material. A givenmicrostructural state is a point that lies along a path ofmicrostructural evolution. Possible paths evolving from astate are circumscribed by the thermodynamics of the alloysystem reflected mostly in its phase diagram. The specificmicrostructural path from some initial state is then deter-mined by the processing applied to the system. The geo-metric change that describes this path is described by acombination of topological events1 and the motion of inter-faces that delineate the constituents in the microstructure.The kinematics of microstructural change relates rates ofchange of the geometric properties in the microstructuralstate to the distribution of interface velocities. Finally, therates of topological processes and the distribution of inter-face velocities are determined by the kinetics of the con-trolling processes. These four fields of scientific endeavor,- geometry, thermodynamics, kinematics and kinetics, -must be combined in order to achieve a sophisticated levelof understanding of the connection of processing to struc-ture in materials science.

2. Geometry and the Microstructural State

Microstructures are three-dimensional space-filling,not-regular, not-random distributions of phases and theirboundaries. The geometric state of a microstructure may bereported at three levels2:

1. The qualitative microstructural state;

2. The quantitative microstructural state; and

3. The topographic microstructural state.

112 DeHoff Materials Research

2.1. The qualitative microstructural state

A microstructure is a three dimensional entity; featuresin the microstructure may have 3, 2, 1, or zero dimensions.That is, features may be volumes (grains, particles), sur-faces (grain boundaries, interfaces), lines (triple lines,edges) or points (quadruple points, vertices on polyhedralparticles). The qualitative microstructural state for a micro-structure is comprised of a list of all of the phases that itcontains (with their compositions and crystal structures)together with a list of all of the classes of 3, 2, 1 and 0dimensional features that are found in the structure. Forexample, Fig. 2 sketches a microstructure with particles ofa β phase distributed in the grain boundary network of theα grains. This structure has α and β-type volumes (the threedimensional phases); αα and αβ interfaces, ααα, ααβtriple lines and αααα, αααβ quadruple points2. The nickelbased superalloy structure sketched in Fig. 3 has γ and γ’phase in the volumes of the grains and oxides, carbides andborides along the grain boundary. A complete feature listfor this system containing all of the classes of interfacesand triple lines has over 30 entries.

The specification of the qualitative microstructural statemay be further embellished with descriptive words that callto mind familiar shapes like platelet, acicular, spheroidal,dendritic, lamellar, etc., or suggest other aspects of thestructure like sparse or space filling.

2.2. The quantitative microstructural state

Each of the features in the qualitative microstructuralstate has associated with it one or more geometric proper-ties. These properties may be defined for individual fea-tures in a collection. They also have global counterpartsthat describe a complete collection of features. For exam-ple, in a collection of particles of a β phase distributed in astructure, each particle has a value of its volume. Thecomplete set of β particles also has a value for its volume.Specification of the quantitative microstructural state isinitiated when one of its geometric properties is measured.

Volume is a familiar example of a geometric propertythat has unambiguous meaning for real three dimensional

features regardless of their shape, size or size distribution.There exists a collection of geometric properties that maybe thought of as fundamental because they have this attrib-ute. Indeed, it is no accident that the geometric propertiesthat are accessible to measurement through the methods ofstereology are precisely those that have general and unam-biguous meaning for real three dimensional microstruc-tures. These fundamental properties fall into the followingcategories:

1. Topological properties;2. Metric properties; a. Measures of extent; b. Curvature measures.Properties are usually reported in a normalized format.

For example, the volume of particles of a set of features isusually reported as the volume of those features per unitvolume of structure. Thus, the measure of extent of threedimensional features is reported as the volume fraction,written VV, occupied by the feature set. These normalizedgeometric properties are listed in Table 1, along with thenotation used to describe each property value.

If a property of a feature (be it 0, 1, 2 or 3-dimensional)is unchanged if the feature is stretched or distorted in acompletely arbitrary way, so long as it is not severed orjoined to itself in the deformation, then that property is atopological property. The most familiar topological prop-erty is the number, N, of disconnected parts of a feature. Itis easy to visualize the number of (three dimensional)grains in a polycrystal (Not so easy to measure it!), or thenumber of (two dimensional) grain faces, or the number ofgrain edge segments. Connectivity, C, reports the numberof cuts that can be made through the feature without sepa-rating it into two parts3. The pore phase in a lightly sintered

Vol. 2, No. 3, 1999 Engineering of Microstructures 113

Figure 2. Microstructure with β phase particles distributed on the grain

boundary network in the α phase, displaying αα and αβ interfaces, and

ααα and ααβ triple lines.Figure 3. A microstructure for a γ/γ’ superalloy with oxides, borides andcarbides at the grain boundaries has a very long feature list.

powder stack has a very high value of connectivity4, de-pending on the particle size and the complexity of thepowder particles. Its connectivity decreases with densifica-tion. The grain edge network in a polycrystal has a highconnectivity. Topological properties provide rudimentaryinformation about the skeleton of the phases in a micro-structure.

Metric properties have values that depend explicitlyupon the shape and size of the features being described.There are two subcategories: measures of extent and cur-vature measures.

The metric measures of extent are the most familiar ofall fundamental geometric properties. They report a quan-titative measure of how much of the feature exists in thestructure. Volume, V, reports this for three dimensionalfeatures, surface area, S, reports it for two dimensionalfeatures and length, L, reports it for one dimensional fea-tures. These are the geometric properties of microstructuresthat are most often quantified. Because these concepts arefamiliar, measured results are most easily interpreted.

Metric curvature measures have a more sophisticatedmeaning, but are indeed fundamental in the sense that theyhave values that provide unambiguous meaning for fea-tures of arbitrary geometry. Curvature is a local propertythat is defined at a point on a surface, or on a space curve.The curvature measures that are accessible are global prop-erties obtained by summing (integrating) the local valuesof curvature over the feature5. Total curvatures may thusbe defined for surfaces5-9 (by integrating the local meancurvatures over the area of the surface to produce theintegral mean curvature, M) and lines10 (by integrating thelocal line curvature along the line, K). The sophisticatedcontent of these concepts makes difficult the visualizationof the meaning of the resulting numbers except in limitingcases. For example, the integral mean curvature, M, of acollection of platelets or flakes reports the total length ofthe edges of those objects8,9. The fact that these globalproperties are fundamental in the sense described in this

section, and that they may be measured using simplestereological methods makes their study important.

Averages of these properties may be defined by com-bining them as shown in Table 2. For example, take anyglobal measure of extent and divide by the number offeatures to produce the number weighted average of thatproperty. The average volume of particles in an array is justV/N; the average surface area is S/N. Other average meas-ures may be defined.

The mean lineal intercept of features <λ>11,12 (theaverage surface-to-surface distance through features in thestructure, Fig. 4, is widely used to characterize the scale ofthose features in a microstructure. This property is funda-mental and is given by

<λ> = 4VS

This measure of scale must be used knowledgeably. Forexample, for platelets <λ> is twice their average thickness,but provides no information about the scale of the structurein the plane of the plates8. For rods, <λ> is twice the averagediameter, but contains no information about the average rod


Table 1. Feature list table: three dimensional structures.

Dim. Description One phase Two phases Three phases

3 Volume α α,β α,β,γ

2 Surface αα αα,αβ,ββ αα,αβ,αγ,ββ,βγ,γγ

1 Lines ααα ααα,ααβ,αβββββ

ααα,ααβ,ααγ,αββ,αβγ,βββ,ββγ,βγγ

γγγ

0 Points αααα αααα,αααβααββ,αβββ,

ββββ

αααα,αααβ,αααγ,ααββ,ααβγ,ααγγ,αβββ,αββγ,αβγγ,αγγγ,ββββ,βββγββγγ,βγγγ,γγγγ

Table 2. Geometric properties of feature sets.

Features Geometric Property Symbol/Units

TopologicalProperties

α,β... Number density NV (m-3)

α,β... Connectivitydensity

CV (m-3)

Metric properties

α,β... Volume fraction VV (m3/m3)

αα,αβ... Surface area density SV (m2/m3)

αα,αβ... Integral meancurvature

MV (m/m3)

ααα,ααp... Length density LV (m/m3)

length8. The mean lineal intercept is the most popularmeasure of grain size in polycrystals.

The average mean surface curvature7, <H>, is given by

< H > = MS

This property becomes important in structures in whichcapillarity effects play a key role, since the local meancurvature is the geometric property at the focus of thethermodynamic behavior of curved surfaces13. The dihe-dral angle at edges in a structure also plays a key role incapillarity driven behavior in microstructures. The averagedihedral angle, <χ> is a fundamental property that may bedefined and measured10.

The distribution of geometric property values within afeature set may be visualized in terms of fundamentalvariables, since these properties can be defined for each

feature in the set. Quantitative measurement of the distri-bution of geometric properties over a set of features in amicrostructure presents a significant challenge. Methodsavailable for estimating size distributions are limited tovery simple structures in which all of the features share thesame quantitative shape.

2.3. The topographic microstructural state

Concepts in the quantitative microstructural state en-compass properties of individual features and the globalproperties of feature sets. They provide no informationabout the spatial distribution of features in the microstruc-ture; this class of information is specified in the topographi-cal microstructural state. There are three primary categoriesof concepts in this level of description:

1. Gradients, which report systematic variations ofgeometric properties with position in the structure;

2. Anisotropies, which report systematic variations withorientation;

3. Associations, which report information about theposition of features in the structure with respect to eachother.

Gradients in microstructural properties typically arisewhen processing imposes a gradient in the influences liketemperature, state of stress, atmosphere, and the like thatoperate to change the microstructure, Fig. 5a. Perhaps themost common example is provided in the surface treatmentof materials; e.g., carburization or nitriding produce obvi-ous microstructural gradients from the surface inward. Ifthe global properties of the structure are of interest, then thepresence of a gradient is a stereological sampling problemthat must be properly averaged. Alternatively, the gradientmay be quantified by taking measurements systematicallyat points along it.

Figure 4. Illustration of the meaning of various measures of <λ>, the

mean lineal intercept in a two phase microstructure structure: (a) λβ is the

mean lineal intercept of the β phase; (b) λα is the mean intercept of the αphase (the mean free path between β particles); (c) λgrains is the meanintercept of a grains.

Figure 5. Microstructures illustrating the concepts of (a) gradients, (b)anisotropies, and (c) associations.

Table 2.Features you see on a section through the microstructure.

Feature in 3-D Designation Feature in 2-D

VolumeGrainParticle

αβ

AreaGrain (section)

Particle (section)

SurfaceGrain BoundaryInterface

αα,ββαβ

LineGrain boundary (trace)Phase boundary (trace)

Line (space curve)Grain edge (tripleline)Triple lines

αααααβ,αββ

PointTriple pointTriple point

PointQuadruple pointQuadruple points

αααααααβααββαβββββββ

Not observed " " " " " " " "


Anisotropies in the distribution of microstructural prop-erties also arise from nonuniformities of the influencesapplied during processing, Fig. 5b. The most obvious ex-ample is the anisotropy of surface area in grain structuresthat arises during directional mechanical processing. An-isotropies also are common in solidification processing. Asin the case of gradients, if the global properties of thestructure are of interest, (e.g., the total surface area of grainsin a deformed polycrystal) then the anisotropy presentsitself as a sampling problem that must be appropriatelyaveraged14,15. On the other hand, the nature of the anisot-ropy may be characterized quantitatively by taking meas-urements in different directions16.

The concept of geometric associations may be appliedto features from the same feature set, or alternatively, tofeatures from different sets in the structure, Fig. 5c. Acollection of particles of the β phase may exhibit departurefrom a random spatial distribution by showing a tendencyto cluster together (forming clumps in the limiting case) orto order (forming a regular lattice of particles like the γ’ordered phase common in many in nickel based superal-loys). These tendencies may in principle be quantified bymeasuring the distribution of nearest neighbor distances.Such measurements have been applied on two dimensionalsections17,18. Undertaking such measurements in three di-mensions requires a reconstruction of the three dimensionalstructure from serial sections, confocal microscopy, acous-tical microscopy or tomography17. Such approaches are notpractical for opaque solid materials, where the only feasibleapproach, serial sectioning, requires an inordinate invest-ment of effort4.

Features in one feature set may tend to associate withthose of another in a microstructure, or to avoid them. Anobvious example is shown in Fig. 5d where the oxideparticles decorate the grain boundaries. Particles may tendto cluster on, or avoid, the grain edges in the structure.Measures of these tendencies are contained in contiguitymeasures19-21. These take the form of fractions of the valueof the measure of extent of one feature set that is due to theassociation of another feature set. Consider for example athree phase structure, αβγ. One measure of the tendencyfor the γ phase to be associated with the α phase is the ratioof αγ surface area to the total boundary area of α particles,(αβ + αγ). The fraction of the length of grain edges in theα matrix occupied by β particles is a measure of thetendency for β particles to associate with grain edges. Othermeasures incorporate triple line lengths.

All of the concepts discussed in this section are funda-mental in the sense that they have unambiguous meaningfor features in arbitrary real three dimensional microstruc-tures. All may be experimentally measured by applying themethods of stereology3,11,12. With a set of properly de-signed samples from probe populations each of the funda-

mental geometric properties listed in Table 1 may be esti-mated with statistical precision. For each geometric prop-erty, separate measurements may be made for all of thefeature sets in the qualitative microstructural state that havethat property. For example, for the structure shown in Fig.6, all of the properties: Vv

α , Vvβ, Sv

αα, Svαβ, Sv

ββ, Lvααα,

Lvααβ, Lv

αββ and Lvβββ , may be stereologically estimated.

3. Thermodynamics in MicrostructuralDesign

Classical phenomenological thermodynamics providesthe basis for organizing information about the behavior ofmatter. The laws of thermodynamics lead to a generalcriterion for equilibrium expressed as an extremum princi-ple: “the entropy of an isolated system is a maximum atequilibrium”13. This extremum principle is the basis for ageneral strategy for finding conditions for equilibrium inthe most complex kind of system. These conditions forequilibrium are sets of equations that must be obeyed inorder for a system to be in its equilibrium state. Variablesin these equations are specific properties of the system:temperatures, pressures, chemical potentials. Definitionsfor these properties, and the associated means for theirmeasurement, arise in the development of the apparatus fordescribing the complex kinds of systems that are common-place in materials science.

In its application to materials science the goal of ther-modynamics is the calculation of equilibrium states. If onetakes system I in state A and places it in environment E it

Figure 6. Two phase microstructure exhibiting all of the features in the

two phase feature list: α, β, αα, αβ, ββ, ααα, ααβ, αββ, βββ.


will change toward a final, equilibrium condition predictedto be system II with state B. In phenomenological thermo-dynamics this goal has been realized for very generalclasses of systems: multicomponent, multiphase, open, re-acting systems in which capillarity effects, crystal defects,elastic effects and external fields may play a role13. In thefollowing sections discussion is limited to the two majorapplication of thermodynamics in the engineering of mi-crostructures:

1. Thermodynamics of phase diagrams; and2. Thermodynamics of capillarity effects.Phase diagrams provide the context within which proc-

esses occur and microstructures develop. Capillarity effectsexplain how the thermodynamics of the phase diagramoperates through the geometry of the microstructure.

3.1. Thermodynamics of phase diagrams

The collection of equilibrium states for any system isrepresented as a map of the domains of stability of thevarious phase forms that a system may exhibit in a spacethat represents ranges of interest of the fundamental over-arching variables, temperature, pressure and composition.Phase diagrams, once determined by direct observation ofthe phases displayed in a survey of combinations of com-positions and temperatures for a system, are now derivedfrom an integrated iteration of thermodynamic informationand direct observation22. Further, first principle calcula-tions of phase diagrams from phase stability and solutionformation behavior derived from interactions modeled onthe atomic scale23 are becoming increasingly successful.

As mentioned before, phase diagrams play a central rolein the engineering of microstructures because they providethe context within which processes occur. In any attemptto understand what happened, or to predict what will hap-pen, when microstructure I was subjected to treatment Eand became microstructure II the first step should be to lookat the phase diagram. However, phase diagrams presentpatterns of equilibrium states in temperature-pressure-composition space. They provide information only aboutthe ultimate state toward which a system tends. A phasediagram does not provide the path that the system willfollow in its pursuit of equilibrium, although in simplecases the path may be strongly indicated. The rate at whichthe path is traversed by the system is also beyond theapplication of phase diagrams. The path, the competingkinetic processes that determine it, and its associated raterequire another level of detail and sophistication in termsof the information required. Kinematic and kinetic consid-erations in microstructural design are treated in more detailin later sections.

The modern trend for constructing phase diagrams bycombining direct structural observations with thermody-namic data and first principle modeling is resulting in a toolfor engineering of microstructures with perhaps unforseen

power. An early objective in these developments was toprovide a means for using all of the data available for agiven system in order to construct an optimal, reliable andconsistent phase diagram for that system22. A second ob-jective was the use of models to predict behavior in ternaryand higher order systems by extending successful modelsfor the binary systems involved. The investment requiredto produce experimental phase diagram information forquaternary and higher order systems makes such an en-deavor impractical. Yet many materials of crucial techno-logical importance are truly “multicomponent”. It seemslikely that systematic studies of such higher order systemsmust rely heavily upon theoretically calculated phase dia-grams.

Perhaps the greatest potential that will be derived fromthis marriage of theory and experiment will be the applica-tion of the thermodynamic information underlying a suc-cessfully constructed phase diagram. The free energy /composition / temperature / pressure relationships that de-scribe the energetics that determine the pattern of winnersin the competition to determine domains of phase stabilityconstitute information that is invaluable to an under-standing of the kinetic processes that compete. In the firstinstance this information also determines the competitionfor the metastable phases that may exist, and provide quan-titative information about how metastable these phasesmay be. The variation of free energy difference betweencompeting phases with composition and temperature pro-vides the pattern of “driving forces” that play a key role inmost kinetic processes. This kind of quantitative energeticinformation is only now beginning to become available forsystems beyond the binary. This is a case in which theinformation being developed has the potential to exceedour abilities to make use of it until we learn how to applyit in multicomponent systems.

The strategies and procedures developed for computingphase diagrams from interactions at the atomic level pro-vides access to kinds of information not experimentallyavailable. For example by incorporating elastic stresses andstrains into the thermodynamic models it is possible todevelop “coherent phase diagrams”23. In processes thatinvolve ordering or spinodal decomposition the earlystages produce particles that are coherent with the sur-rounding crystal lattice, i.e., the phase that forms and theparent phase share the same crystal lattice. In this casesignificant elastic energy is developed in accommodatingthe phases and the predicted domains of stability may besignificantly different from those observed in bulk systems.These elastic effects must be incorporated into the evalu-ation of the driving forces for the kinetic processes thatoccur.


3.2. Thermodynamics of capillarity effects

A microstructure is a spatial distribution of geometricentities: interfaces, triple lines, quadruple points. The sub-set of the thermodynamic apparatus that incorporates thesegeometric entities is called capillarity. The conditions forequilibrium involving geometric aspects of microstructurefall into two main scenarios:

1. Equilibrium at a curved interface where the pressuredifference across the interface is proportional to the surfacefree energy and the local mean curvature, Fig. 7; and

2. Equilibrium at triple lines where a balance of theenergies of the three meeting surfaces determines the localconfiguration, Fig. 8.

Both of these effects play important roles in determin-ing microstructural evolution and thus in engineering mi-crostructures.

The mechanical effect described in the first case can beshown to translate into shifts in phase boundaries on phase

diagrams. These translations then operate to influenceprocesses involving those phases. The thermodynamic ef-fects appear as an increase in vapor pressure for curvedliquid/vapor interfaces, a decrease in the melting point forcurved solid/liquid interfaces, shifts in vacancy concentra-tions for crystal/vapor interfaces, or shifts in compositionfor solid/solid interfaces involving two or more compo-nents. These "capillarity shifts" may act as the driving forcefor processes, as in coarsening, grain growth or sintering,may modify the driving force as in diffusion controlledgrowth, or may act as a barrier to be overcome, as innucleation. These microstructural processes cannot be de-scribed without invoking the thermodynamics of capillar-ity.

Equilibrium at triple lines tends to have an obviouseffect on microstructural geometry. The dihedral anglesthat develop in balancing the surface energies of the meet-ing interfaces determine “wetting tendencies”. These as-pects of the thermodynamics of interfaces appear insoldering and joining (where wetting is desirable) and inliquid phase penetration (where it is to be avoided). Theyplay a crucial role in nucleation processes, favoring hetero-geneous nucleation at grain boundaries, triple lines or quad-ruple points, or on dispersed inoculants or container walls.

Capillarity thermodynamics and the energetics of phasediagrams are key components in the quest to engineermicrostructures. We are only now beginning to understandthe operation of these parts of the thermodynamic appara-tus in multicomponent systems.

4. Kinematics and the Path ofMicrostructural Change

In mechanics kinematics means the description of thegeometry of motion (as in the motion of a compound leversystem) without regard to the physical influences that causethe motion. For example in kinematics it is shown that thecurve traced out by a point on a wheel as the wheel rollsalong a line is a cycloid. The time variation of the spatialdistribution of the components in a mechanical system maybe described in terms of velocities of specific components,e.g., the displacement of a pin in the mechanism, or theangular rotation velocity of a lever.

4.1. The kinematic equations

There also exists a kinematics of microstructural evo-lution. In this case, the motions are visualized as the distri-bution of velocities over a moving interphase interface, Fig.9. These motions produce changes in the geometry of thephases involved. A set of kinematic equations relatechanges in the geometric properties of the microstructureto this velocity distribution.

Where V, S and M are the volume, surface area andintegral mean curvature of the phase bounded by the mov-

Figure 7. The condition for local equilibrium across a curved αβ interface

requires that the pressure in the β phase be different from the pressure in

the a phase by the factor 2γH where γ is the surface free energy and H isthe local mean curvature of the surface. The chemical potencial of any

component, µk, is the same on both sides of the interface.

Figure 8. The condition for equilibrium at a triple line where threesurfaces meet (a) determines the angle between the surfaces (b). Theseangles are measures of the tendencies for the phases to wet each other andmay strongly influence the microstructure.


ing interface. H is the local mean curvature and K theGaussian curvature of an element of boundary moving withvelocity v. These equations make no simplifying assump-tions about the geometry of the structure3.

dVdt

= ∫ S∫ v dS

dSdt

= 2 ∫∫ S

v H dS

dMdt

= ∫∫ v K dS

The path of microstructural evolution is visualized asthe sequence of geometric states that a system exhibitsduring a microstructural process. The path may be madequantitative by measuring the geometric properties of themicrostructure at a sequence of observations that samplethe process. An individual quantitative microstructuralstate may be visualized as a point in a space that has anumber of dimensions equal to the number of geometricparameters that have been quantified. The path is then aspace curve in that multidimensional space, connectingsubsequent states along the process. The path providesrudimentary information about the process that is occurringthat is not available from observations of the time variationof individual geometric properties.

If the boundary motions are smooth and continuoussome information about what determines the path may beobtained from ratios of the kinematic equations:

dSdV

=

2 ∫∫ S

v H dS

∫∫ S

vdS

dMdV

=

∫∫ S

v K dS

∫∫ S

v dS

The sequence of geometric states, expressed in terms ofhow the surface area and integral mean curvature vary withthe volume of the phase, is seen to be determined in acomplex way by ratios of integrals over the area of themoving interface of weighted measures of the velocitydistribution on the moving interface.

4.2. Topological processes in microstructural evolution

Most microstructural paths result from two classes oflocal processes:

1. The smooth migration of interfacial elements underthe control of local atomic accommodations. These pro-duce changes in the metric properties of the structure andare governed by the kinematic equations.

2. Discontinuous changes in the structure that alter thetopology of the microstructure. The time frame for thesechanges is short compared with those associated withsmooth boundary migration.

Smooth boundary migrations produce changes that areglobal because they act at every point on the movinginterface. Topological processes are by their nature local,affecting only elements of the microstructure in their im-mediate area. Both classes of processes contribute signifi-cantly to determining the path of microstructural change.

The coupled operation of topological and metric proc-esses is a common phenomenon in microstructural evolu-tion. Typically the structure changes smoothly as interfacesmigrate; this migration sets up a local shape change thatreaches a critical condition and, on a short time scale, thetopological process occurs. Some familiar examples fol-low.

Grain growth results from the disappearance of grainsin a space filling polycrystal, a topological process whichdecreases the number of grains in the microstructure. (Asecond topological process in which grains switch neigh-bors is responsible for maintaining the distribution of graintopologies in the system.) The geometric configurationnecessary for these topological events to occur is set up bythe smooth and continuous migration of grain boundariesuntil locally some critical condition has been attained anda topological change occurs.

Sintering is characterized by topological changes thatdisconnect the pore network until pores become isolated; if

Figure 9. The distribution of velocities over the interface that bounds afeature illustrated in this figure is related to the changes in the geometricproperties of the feature through the kinematic equations.


the microstructural context is correct, these isolated poresthen disappear. These topological events are set up by thesmooth migration of pore-solid interface until some localcritical geometric condition is attained.

Coarsening involves smooth boundary migration, withinterfaces bounding large particles moving outward andthose bounding small particles moving inward. The latterultimately results in a topological process, the disappear-ance of small particles. As a result, the number of particlesdecreases and the average particle size increases.

Precipitation begins with a topological process, theformation of new particles in nucleation. Further micros-tructural development derives from growth, which is thesmooth and continuous migration of the surface boundingprecipitate particles.

The topological processes are by their nature instanta-neous. Topological events are disruptions in the otherwisesmooth evolution of the geometry of the structure. A clusterof atoms attains a critical size and a new particle is definedto be nucleated. A small grain shrinks inward on itself untilit disappears in grain growth or coarsening. A channel in apore network reaches a critical configuration and at someinstant in time pinches closed, further disconnecting thepore network. Rates of change of local values of the metricproperties may be large in the vicinity of a topological eventimmediately before or after it occurs. However, globalmetric properties change smoothly and continuously asthey average over contributions from these events.

Qualitatively, the path is characteristic for a given classof process (e.g., grain growth, coarsening, etc.) The quan-titative path is then determined by the choices of processingvariables applied to the system. For example, in precipita-tion increasing the supersaturation may increase the nu-cleation rate relative to the growth rate and alter the path toproduce a structure with a larger number of particles thatare on the average smaller in size. The precipitation processthus exhibits an envelope of paths. This envelope of pathsdisplays the ensemble of microstructural states that areaccessible to the process. The choice of the processingvariables selects a specific path from this envelope of pathsand narrows the focus to microstructural states along thatpath. Interruption of the process at the appropriate timeleaves the structure in a selected microstructural state.

5. The Kinetics of Microstructural EvolutionThe kinematic equations describe the global geometric

changes of a microstructure in terms of the distribution ofinterface velocities. The kinematic equations become kineticequations when the interface velocity function v is evaluatedfrom physical considerations including the thermodynamiccontext and the atomic processes that make the interfacemove. In general the local velocity of an element of movinginterface is governed physically by an interface accommoda-tion equation24,25. A phase boundary exists because the two

phases that define it have different properties. In order forthe boundary to move, these differences must be accommo-dated. This is generally achieved through local flows ofchemical components (to accommodate composition dif-ferences) or heat (to accommodate enthalpy differences).

To illustrate the kinetic evaluation of the local velocity v,consider the case of a boundary between two binary phases,α and β, that have two different compositions at the interface,Cα and Cβ, Fig. 10. In order for the interface to move thisconcentration difference must be accommodated by supply-ing (or removing) atoms of the components. In this illustrationlet it be assumed that the motion of the interface is controlledby the supply of these atoms by diffusion in the adjacentphases. If the process is controlled by this diffusion then localequilibrium obtains at the interface and Cα and Cβ are molarconcentrations of the component (say component B) com-puted from the equilibrium tie line for α and β at the tempera-ture of the interface in the phase diagram for the system.Define Jα and Jβ to be the diffusion fluxes of component B(moles of B/cm2 -sec) at the interface in the α and β phasesrespectively. Then it can be shown24,25 that conservation ofcomponent B requires that

v = Jβ − Jα

Cβ − Cα

This is the compositional interface accommodationequation.

5.1. Precipitation processes

An alloy with composition Co is first solution treated ata high temperature where the phase diagram, Fig. 11, showsa single phase to be the equilibrium structure. This alloy is

Figure 10. Illustration of the factors involved in the interface accommo-dation equation that describes the velocity v of a phase of the boundarybetween two phases.


then quenched to an aging temperature that lies within atwo phase α + β field in the phase diagram. Particles of theβ phase nucleate and begin to grow. Typically the β parti-cles form and grow at a composition that is near theirequilibrium composition, Cβ. With this condition, there isno gradient in the β phase during growth: Jβ = 0, so that

v = − Jα

Cβ − Cα

The flux at the interface in the α phase can be expressedin terms of the concentration gradient at the interface, ∇C,and the diffusion coefficient in the α phase:

Jα = − D ∆ C

If the process is controlled by diffusion in the matrixphase then a concentration field will evolve with time inthe matrix. This composition distribution is a solution toFick’s second law which governs diffusion determined bythe initial conditions and the boundary conditions at the αβinterface. This concentration field will exhibit flux linesthat connect to the particle surfaces. A trace of the compo-sition distribution along such a flux line is shown in Fig.12. The gradient at the interface may be expressed in termsof the composition difference (Co - Cα) and a local diffusionlength scale, λ, shown in Fig. 12.

∆ C = C0 − Cα

λ

The diffusion length scale for any element of interfacegrows parabolically with the time since the particle nucle-ated, until concentration interference occurs in the diffu-

sion fields between neighboring particles. Thereafter thegrowth rate of the length scale slows down. The growth rateapproaches zero as the length scales approach infinity andthe system approaches equilibrium. The distribution ofdiffusion length scales at any time in this process is tied tothe time dependence of the nucleation rate since the lengthscale for each surface element starts from zero at its mo-ment of nucleation.

The interface accommodation equation for diffusioncontrolled growth can thus be written

v = D

Cβ − Cα C0 − Cα

λ = D [

C0 −Cα

Cβ − Cα ] 1λ

The factor in brackets, which is the ratio of the compo-sition difference in the matrix phase to the compositiondifference across the phase boundary, is called the super-saturation ratio for the process, and can be determinedfrom the tie line on the phase diagram that describes thistwo phase system. D is the diffusivity in the matrix phaseat the composition Cα, and λ is the local diffusion lengthscale in the matrix at a surface element.

In a microstructure that is evolving by diffusion con-trolled growth of a β phase into an α matrix the interface

velocity, v, is expected to vary with position over the αβinterface. Inspection of Eq. (9) reveals that all of thatvariation is embodied in the diffusion length scale, λ. Thus,the distribution of diffusion length scales in the structureplays a central role in determining the path and kinetics ofthe evolution of the microstructure.

Rates of change of the global geometric properties canbe related to the local physics embodied in Eq. (9) byinserting this expression for the interface velocity into thekinematic equations. The rate of change of the volumeof the growing β phase is given by

Figure 12. (a) Two dimensional sketch of a diffusion generated concen-

tration field for β precipitating in α, showing the isoconcentration con-

tours and flux lines. (b) For any point P on the αβ interface the

concentration gradient in the α along the flux line can be described by a

composition difference (Ck0 - Ck

α) and a diffusion lenght scale, λ.

Figure 11. Typical phase diagram in which a precipitaion process can bemade to occur.


change of the global geometric properties can be related tothe local physics embodied in Eq. (9) by inserting thisexpression for the interface velocity into the kinematicequations. The rate of change of the volume of thegrowing β phase is given by

dVdt

= ∫∫ S

v dS = ∫∫ S

D [ C0 − Cα

Cβ − Cα ] 1λ

dS =

D [ C0 − Cα

Cβ − Cα ] ∫∫ S

1λ

dS

dVdt

= D [ C0 − Cα

Cβ − Cα ] < 1λ

> S S

where < 1λ

> S is a surface area weighted harmonic mean

of the distribution of diffusion length scales. The kineticsof growth of the β phase is clearly determined by thetemperature of the process through the diffusioncoefficient in the α phase, which is strongly temperaturesensitive, and the supersaturation ratio obtained from thephase diagram, which may not be as sensitive totemperature. The nucleation part of the precipitationprocess is implicit in the distribution of diffusion lengthscales. If nucleation is in the site saturation mode (all ofthe available nucleation sites are activated at t = 0) thenthe distribution of diffusion length scales will be verynarrow, at least until the late stages of the process whereconcentration interference becomes dominant. Asnucleation becomes time dependent, this distributionbroadens.

The time dependence of surface area and integral meancurvature can be determined by inserting the velocity Eq.(9) into their kinematic Eqs. (3):

dSdt

= 2 D [ C0 − Cα

Cβ − Cα ] < 1λ

> M M

and, for the change in M

dMdt

= D [ C0 − Cα

Cβ − Cα ] < 1λ

> Ω Ω

Insight into the path of microstructural change for amicrostructure evolving by diffusion controlled growth ofthe β phase can be obtained by taking ratios of these kineticequations. The projection of the path which plots thesurface area of the microstructure versus the volume of thegrowing phase is given by the ratio

dSdV

=

2 D [ C0 − Cα

Cβ − Cα ] < 1λ

> M M

D [ C0 − Cα

Cβ − Cα ] < 1λ

> S S = 2

< 1λ

> M M

< 1λ

> S S

The component of the path represented by the variationof the integral mean curvature with the volume of thegrowing particles is

dMdV

=

D [ C0 − Cα

Cβ − Cα ] < 1λ

> Ω Ω

D [ C0 − Cα

Cβ − Cα ] < 1λ

> S S =

< 1λ

> Ω Ω

< 1λ

> S S

In both of these equations the dependence upon thediffusion coefficient in the matrix, D, and the phase dia-gram concentrations contained in the supersaturation ratiocancel. The path depends upon the ratio of differentlyweighted averages of the diffusion length scale distribu-tion, and current values of geometric properties, S, M andΩ. The breadth of the diffusion length scale distribution isdetermined by the time dependence of the nucleation be-havior in the process, i.e., by the rate of topological change.It is concluded that the variability of the path of microstruc-tural evolution in diffusion controlled precipitation islargely determined by the kinetics of its topological nuclea-tion process: nucleation

All of these results reflect the microstructology view-point: the description is carried as far as possible beforeintroducing simplifying geometric assumptions.

In order to apply the kinematic equations to specificmicrostructural processes it is necessary to visualize thedistribution of concentration gradients in the matrix at themoving interface. This evaluation is specific to each classof process.

5.2. Diffusion controlled coarsening

Coarsening sets in when the chemical driving force forgrowth of a precipitate phase has been exhausted so that themuch smaller differences in composition associated withcapillarity effects may begin to operate. In this case thethermodynamics of capillarity effects predicts that the con-centration in the α phase at a surface element is determinedby the local mean curvature of the element13. An elementof surface on a small particle has a relatively high concen-tration in the adjacent α phase; surface elements on coarseparticles have a relatively small concentration in the adja-cent α phase. Thus, there exists a distribution of concentra-tions over the collection of boundaries of the α phase. If theprocess is controlled by diffusion between β particlesthrough the matrix α phase, then a concentration fieldevolves with time in this process. This evolution is gov-erned by Fick’s second law and the distribution of concen-trations at the αβ surfaces that bound the α phase.

The composition field will exhibit flux lines (normal tothe isocomposition contours) that connect any given sur-face element with a communicating neighbor element. Fig.13 sketches a composition profile along such a flux line.


The gradient at the interface for the smaller particle can bevisualized in terms of the composition difference betweenthe surface elements (which is determined by the differencein their mean curvatures) and the diffusion length scale, λ:

∆C = C(H) − C(H)n

λ =

2 γ Vβ

R T

Hn − Hλ

In this result γ is the surface free energy of the interface,Vβ is the molar volume of the β phase, R is the gas constant,T the absolute temperature Hn the mean curvature of thecommunicating neighbor element and H that of the refer-ence element. Insert this evaluation into the interface ac-commodation equation:

v = D

Cβ − Cα ∆C = D

Cβ − Cα 2 γ Vβ

R T

Hn − Hλ

v = KD Hn − H

λ

Note that if H is larger than Hn (implying the referenceelement is on a smaller particle) v is negative and theparticle shrinks. For the opposite case, the particle grows.Thus, in diffusion controlled coarsening the local velocityof an element is determined by its curvature, H, that of itscommunicating neighbor, Hn, and a diffusion length scale,λ which is of the order of the distance between particles.

Insertion of this result into the kinematic equationsgives the predictions for the time evolution of global geo-metric properties for diffusion controlled coarsening. Forthis process the total volume of the β phase does not changeso that dV/dt = 0. It can be shown that this condition yields

the result that the average curvature of the communicatingneighbor elements is simply the average mean surfacecurvature in the system <H> defined in Eq. (2). Insertionof this result into the remaining kinematic equations yieldspredictions of the rate of change of surface area and totalcurvature for diffusion controlled coarsening. In this casethe integrations involved lead to weighted averages of themean surface curvature distribution, as well as the distribu-tion of diffusion length scales. These results contain nosimplifying geometric assumptions. For a complete pres-entation of these results see reference26.

The path of microstructural change for coarsening proc-esses is presumed to be highly constrained. Classical theo-ries on the subject suggest that, after some adjustment tothe initial particle size distribution, all coarsening systems(of any given volume fraction) approach a fixed size distri-bution, and thus a fixed path of microstructural evolution.Factors that might alter the path, such as the spatial distri-bution of the particles (whether clustered or ordered) havebeen left unexplored.

5.3. Grain growth

The local process that governs grain growth is grainboundary migration. In contrast with the previously de-scribed processes, grain boundary migration does not in-volve concentration fields, Fick’s second law or diffusionlength scales. The driving force for grain growth is thedecrease in surface energy associated with the eliminationof grain boundary area. This is accomplished if each ele-ment of boundary moves toward its local center of curva-ture. The classic kinetic model, due to von Neuman27 andMullens28, describes grain growth in a two dimensionalgrain structure. They assumed that the velocity of theboundary element is proportional to its local curvature:

where α is a rate constant that depends upon tempera-ture and composition and k is the local curvature of theboundary. This assumption leads to the classical “6 – n”rule: that rate of change of the area of any grain dependsonly upon the number of sides that is has and the rateconstant α. No other geometric factors are involved4. Thecorresponding hypothesis for grain growth in three dimen-sional microstructures is that the velocity is proportional tothe local mean curvature H of the grain boundary.

v = − α k

The von Neumann result and its three dimensionalanalog neglect the topological processes that are an integralpart of grain growth30,31. The mean grain volume, which isdetermined by the number of grains in the structure and noother geometric parameter, can only change as a result ofthe topological event in which a small grain disappears, Fig.14. As small grains are removed from the system new small

Figure 13. Illustration of the concentration profile along a flux line thatjoins two "communicating neighbors" surface elements and the associated

diffusion lenght scale λ in diffusion controlled coarseing.


grains are formed as a result of the topological switchingprocesses illustrated in Fig. 14. In the vicinity of the cata-strophic topological events the surface curvatures are largeand the interface velocities correspondingly high. Thus, ingrain growth most interface elements are migrating slowlytoward their centers of curvature but, at any instant in time,a small fraction of the surface elements that happen to benear a topological event, are moving very rapidly. Thekinematic equations may be used in principle to predictkinetics and the path of microstructural change. However,it is necessary to incorporate the observation that much ofthe change in surface area of the grain boundary networkthat occurs in any time interval is more or less directlyassociated with the topological events that occur in thattime interval. Indeed, in an analysis of grain growth in twodimensions29 it has been estimated that over 90% of thechange in boundary area is a direct result of the topologicalprocesses.

Paths of microstructural change in grain growth rangefrom “normal grain growth” in which the structure coarsenswithout changing the relative width of the grain size distri-bution, to “abnormal grain growth” or “secondary recrys-tallization” where selected grains grow at a rate farexceeding their neighbors. It seems evident that this rangeof paths is governed by the spatial distribution of thetopological events. In normal grain growth the topologicalevents are uniformly distributed in space. In abnormal graingrowth these processes apparently tend to be clustered nearabnormal grains. Contributing factors include textural ef-fects, nonuniformity in the distribution of second phaseparticles or nonuniformity in the developing “normal”grain size distribution. The material and process variablesthat control the path of microstructural change in graingrowth remain a subject for experimental study.

5.4. Sintering

A geometrically general approach to modeling the sin-tering process requires modification of the kinematic equa-tions. In sintering the migration of the pore solid interfaceis not the only contribution to the changes in the geometricproperties. The annihilation of vacancies at grain bounda-ries plays a significant role in densification which alsocontributes to changes in other geometric properties. Thusintegrals of the local velocity over the moving pore solidinterface do not adequately describe the geometric changesthat occur.

When a stack of particles is heated to a significantfraction of their melting point, necks form at points ofcontact. These necks are grain boundaries in the solidphase. The pore phase is a connected network which inter-penetrates the solid connected network. The necks growuntil they impinge, forming triple lines in the developinggrain boundary network. The formation of triple lines isaccompanied by the pinching off of channels in the porenetwork. Eventually a fully connected grain boundary net-work results as the pore network disconnects into isolatedpores.

The thermodynamics of defect structures combineswith capillarity theory to predict that vacancies are gener-ated at the curved pore surface. These vacancies migrate tonearby grain boundaries between particles. As the vacan-cies annihilate at the grain boundaries, the grain centersmove toward each other and the porous aggregate densifies.

The process may be visualized in terms of a space fillingcell structure in which each cell contains one grain and itsassociated porosity32. Each cell face is partly covered bygrain boundary, partly by porosity. Let f be the ratio of cellface area to grain boundary area at any point in the process.When the equivalent of one layer of vacancies is annihi-lated in the grain boundary on a particular face the volumeof the cell is decreased by the removal of a layer one atomthick from the whole cell face. In the early stages ofsintering, when the grain boundary occupies a small frac-

Figure 14. Topological processes that occur in two dimensional graingrowth: (a) the disappearance of small grains, and (b) the switchingprocess.

Figure 15. Illustraiting the progress in terms of the evolution of the poreand grain boundary structures associated with a single grain.


tion of a cell face, annihilation of a layer of vacancies in thegrain boundary carries with it a large volume change, i.e.,the vacancy annihilation process is efficient. In the latestages of the process after porosity is isolated f approaches1 and the volume change of the structure is roughly equalto the volume of vacancies annihilated.

Sintering is an unusual microstructural process in thatit involves two networks with evolutions that are coupled32.Assume for simplicity that the particles are single crystals.At the outset the pore network is completely connected andthe grain boundary network, which is confined to the inter-particle necks, is completely disconnected. As densifica-tion proceeds and pore channels begin to pinch off asgrowing necks impinge on each other, disconnected por-tions of the grain boundary structure begin to connect toform subnetworks . At a still later stage the grain boundarysubnetworks connect up to form a complete network. Thisgrain boundary network is capable of the metric and topo-logical changes that characterize grain growth. In this stageof sintering, which probably characterizes the range from15 to 5% porosity, the grain boundary network may becomeuncoupled from the pore network. The subsequent path ofmicrostructural change is strongly dependent upon thebalance between grain growth and pore evolution becausethe elimination of porosity is most efficient for pores in thegrain boundary network. The attainment of a fully densebody is favored if the pore structure and the grain boundarynetwork remain coupled. If the grain structure coarsens,either normally or abnormally, pores left without contact tothe grain boundary network become stable and resist elimi-nation.

Paths of microstructural evolution in sintering are cir-cumscribed primarily by the geometry of the initial powderstack embodied in the particle size, size distribution, shapesand stacking4. Processing scenarios that involve precom-paction or hot pressing versus loose stack sintering signifi-cantly alter the path in the neck growth and channel closurestages of the process but have less effect in the late stagesof the process where the pores are isolated. In that stage thecoupling between the grain growth process and pore evo-lution determines the path.

6. SummaryEngineering of microstructures requires wise choices

based upon knowledge on the one hand of the relationbetween processing and the microstructural states that maybe developed and, on the other hand, of the relation betweenmicrostructural states that may be achieved and their prop-erties. This presentation has focused on the tools that areneeded to choose a material and design a process to achievea target microstructural state.

Concepts of microstructural state and the path of mi-crostructural change form the basis for the engineering ofmicrostructures. For a given material and initial micros-

tructural state each class of process yields an envelope ofmicrostructural paths. Fundamental geometric conceptsmust be mastered in order to describe microstructural statesand paths qualitatively, then quantitatively.

A knowledge of the thermodynamics that underlie mul-ticomponent, multiphase systems leads to an understandingof phase diagrams for real, complex systems. These mapsof the domains of stability of possible equilibrium states fora system form the context within which microstructuralprocesses occur.

A set of kinematic equations relate the evolution ofmicrostructural geometry to the distribution of velocitieson boundaries in the structure. These geometrically generalrelationships provide a basis for establishing paths of mi-crostructural change and rates of traverse of those paths.

These kinematic equations may be converted to kineticequations by applying the principles of both reversible andirreversible thermodynamics to endow local interface ve-locities with a physical description. This connection withthe physics of the moving atoms is the basis for establishingthe effect that processing sequence and variables have uponthe path and kinetics of microstructural evolution.

Many common microstructural processes have a topo-logical component. The typical topological event is a localchange in connectivity or number of some geometric entityin the structure. It may be accompanied by relatively largechanges in local geometric properties. Rates of topologicalprocesses relative to rates of metric property changes maybe responsible for introducing breadth into the spread ofpossible microstructural paths that a system may choose.

Engineering of microstructure presumes that a targetmicrostructure has been circumscribed which exhibits acombination of properties desired for a particular applica-tion. The target microstructural state must be producedfrom a selected material through processing. It is a particu-lar state obtained by interrupting a microstructural evolu-tion that carries the material along some microstructuralpath. The spectrum of microstructural paths available froma given starting state is determined by the processing se-quence and processing variables. Understanding these con-nections so that the right material may be selected andsubjected to the right process requires mastery of the ge-ometry, thermodynamics, kinematics and kinetics of mi-crostructural evolution.

References

1. Rhines, F.N. Microstructology, Behavior of Micro-structure of Materials. Dr. Riederer-Verlag, GmbH,Stuttgart, Germany, 1886.

2. DeHoff, R.T. Topography of microstructure. Metal-lography, n. 8, p. 71-87, 1975.

3. DeHoff, R.T.; Rhines, F.N. Quantitative Microscopy.McGraw-Hill, New York, NY, p. 291-325, 1968.


4. Aigeltinger, E.H.; DeHoff, R.T. Quantitative determi-nation of topological and metric properties duringsintering of copper. Met. Trans., n. 6, p.1853-1857,1975.

5. DeHoff, R.T.; Rhines, F.N. Quantitative Microscopy.McGraw-Hill, New York, NY, p. 307-316, 1968.

6. Minkowski, H. Math.Ann., n. 57, p. 447-452, 1903.7. DeHoff, R.T. The quantitative estimation of mean

surface curvature. Trans. Met. Soc. AIME, n. 239, p.617-621, 1967.

8. DeHoff, R.T. Geometrical meaning of the integralmean curvature. Microstructural Science Volume 5,Elsevier-North Holland, New York, p. 331-343, 1977.

9. DeHoff, R.T. Integral mean curvature and plateletgrowth. Met. Trans., n. 10A, p. 1948-1949, 1979.

10. DeHoff, R.T.; Gehl, S.M. Quantitative microscopy oflineal features in three dimensions. Proc. 4th Intl.Cong. For Stereology, National Bureau of Standards,Gaithersburg, MD, p. 29-40, 1975.

11. Underwood, E.E. Quantitative Stereology, Addison-Wesley Pub. Co., Reading, MA, p. 41, 1970.

12. Kurzydlowski, K.J.; Ralph, B. The Quantitative De-scription of the Microstructure of Materials, CRCPress, Boca Raton, FL p. 275, 1995.

13. DeHoff, R.T. Thermodynamics in Materials Science.McGraw-Hill, New York, NY, p. 355-383, 1993.

14. Hilliard, J.E. Specification and measurement of mi-crostructural anisotropy. Trans. Met. Soc. AIME, n.224, p. 1201-1207, 1962.

15. Baddeley, A.J.; Gundersen, H.J.G.; Cruz-Orive, L.M.Estimation of surface from vertical sections. J. ofMicroscopy, n. 142, p. 259-271, 1986.

16. Underwood, E.E. Quantitative Stereology, Addison-Wesley Pub. Co., Reading, MA, p. 49-79, 1970.

17. Russ, J.C. Computer Assisted Microscopy. PlenumPress, New York, NY, p. 261-264, 1990.

18. Schwarz, H.; Exner, H.E. The characterization of thearrangement of feature centroids in planes and vol-umes. J. of Microscopy, n. 129, p. 155-169, 1983.

19. Gurland, J. The measurement of grain contiguity intwo phase alloys.Trans. Met. Soc. AIME, n. 212, p.452-457, 1958.

20. Cahn, J.W.; Hilliard, J.E. The measurement of graincontiguity in opaque samples. Trans. Met. Soc. AIME,n. 224, p. 759-765, 1959.

21. Underwood, E.E. Quantitative Stereology, Addison-Wesley Pub. Co., Reading, MA, p. 99-103, 1970.

22. Kattner, U.R. The thermodynamic modeling of mul-ticomponent phase equilibria. Journal of Metals, p. 14- 19, December, 1977.

23. Fontaine, D. de. From Gibbsian thermodynamics toelectronic structure: nonempirical studies of alloyphase equilibria, MRS Bulletin, p. 16 - 25, August,1996.

24. Sekerka, R.F.; Jeanfils, C.L.; Heckel, R.W. Lectureson the Theory of Phase Transformations, Aaronson,H.I., ed., AIME, New York, NY p. 117-132, 1975.

25. Geiger, G.H.; Pourier, D.R. Transport Phenomena inMetallurgy, Addison-Wesley, Reading, MA, p. 491-496, 1973.

26. DeHoff, R.T. A geometrically general theory for dif-fusion controlled coarsening. Acta Metallurgica, n.39, p. 2349-2360, 1991.

27. Neumann, J. von. Curvature driven grain growth intwo dimensions. Metal Interfaces, ASM Seminar,ASM, Materials Park, OH, p.108-118, 1952.

28. Mullins, W.W. Role of grain boundary grooving ingrain growth in two dimensions. J. Appl. Phys., n. 27,p. 900-905, 1956.

29. DeHoff, R.T. Metric and topological contributions tothe rate of change of boundary length in two dimen-sional grain growth. Acta.Mat., in press.

30. Smith, C.S. Grain shapes and other metallurgical ap-plications of topology. Metal Interfaces, AmericanSociety for Metals, Cleveland, OH, p. 95-108, 1952.

31. Craig, K.R.; Rhines, F.N. Mechanisms of steady stategrain growth in aluminum. Met. Trans., n. 5, p. 413-419, 1974.

32. DeHoff, R.T. A Stereological model of sintering. Sci-ence of Sintering. New Directions for Materials Proc-essing and Microstructural Control, Uskokovic, D.P.;Palmour III, H.; Spriggs, R.M., eds., Plenum Press,New York, p. 55-72, 1989.


engineering of microstructures - scielo · engineering of microstructures ... the useful object...

Documents