Sintering is characterized by topological changes that disconnect the pore network until pores become isolated; if the microstructural context is correct, these isolated pores then disappear. These topological events are set up by the smooth migration of pore-solid interface until some local critical geometric condition is attained. Coarsening involves smooth boundary migration, with interfaces bounding large particles moving outward and those bounding small particles moving inward.

The latter ultimately results in a topological process, the disappearance of small particles. As a result, the number of particles decreases and the average particle size increases. Precipitation begins with a topological process, the formation of new particles in nucleation. Further microstructural development derives from growth, which is the smooth and continuous migration of the surface bounding precipitate particles. The topological processes are by their nature instantaneous. Topological events are disruptions in the otherwise smooth evolution of the geometry of the structure. A cluster of atoms attains a critical size and a new particle is defined to be nucleated.

A small grain shrinks inward on itself until it disappears in grain growth or coarsening. A channel in a pore network reaches a critical configuration and at some instant in time pinches closed, further disconnecting the pore network.

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Rates of change of local values of the metric properties may be large in the vicinity of a topological event immediately before or after it occurs. However, global metric properties change smoothly and continuously as they average over contributions from these events. Qualitatively, the path is characteristic for a given class of process e. The quantitative path is then determined by the choices of processing variables applied to the system. For example, in precipitation increasing the supersaturation may increase the nucleation rate relative to the growth rate and alter the path to produce a structure with a larger number of particles that are on the average smaller in size.

The precipitation process thus exhibits an envelope of paths. This envelope of paths displays the ensemble of microstructural states that are accessible to the process. The choice of the processing variables selects a specific path from this envelope of paths and narrows the focus to microstructural states along that path.

Interruption of the process at the appropriate time leaves the structure in a selected microstructural state. The kinematic equations describe the global geometric changes of a microstructure in terms of the distribution of interface velocities. The kinematic equations become kinetic equations when the interface velocity function v is evaluated from physical considerations including the thermodynamic context and the atomic processes that make the interface move. In general the local velocity of an element of moving interface is governed physically by an interface accommodation equation 24, A phase boundary exists because the two phases that define it have different properties.

In order for the boundary to move, these differences must be accommodated. This is generally achieved through local flows of chemical components to accommodate composition differences 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, a and b , that have two different compositions at the interface, C a and C b , Fig. In order for the interface to move this concentration difference must be accommodated by supplying or removing atoms of the components.

In this illustration let it be assumed that the motion of the interface is controlled by the supply of these atoms by diffusion in the adjacent phases. If the process is controlled by this diffusion then local equilibrium obtains at the interface and C a and C b are molar concentrations of the component say component B computed from the equilibrium tie line for a and b at the temperature of the interface in the phase diagram for the system. Then it can be shown 24,25 that conservation of component B requires that.

This is the compositional interface accommodation equation. An alloy with composition C o is first solution treated at a high temperature where the phase diagram, Fig. Particles of the b phase nucleate and begin to grow. Typically the b particles form and grow at a composition that is near their equilibrium composition, C b.

If the process is controlled by diffusion in the matrix phase then a concentration field will evolve with time in the matrix. This composition distribution is a solution to Fick's second law which governs diffusion determined by the initial conditions and the boundary conditions at the ab interface.

This concentration field will exhibit flux lines that connect to the particle surfaces. A trace of the composition distribution along such a flux line is shown in Fig. The gradient at the interface may be expressed in terms of the composition difference C o - C a and a local diffusion length scale, l , shown in Fig. The diffusion length scale for any element of interface grows parabolically with the time since the particle nucleated, until concentration interference occurs in the diffusion fields between neighboring particles.

Thereafter the growth rate of the length scale slows down.

The growth rate approaches zero as the length scales approach infinity and the system approaches equilibrium. The distribution of diffusion length scales at any time in this process is tied to the time dependence of the nucleation rate since the length scale for each surface element starts from zero at its moment of nucleation. The interface accommodation equation for diffusion controlled growth can thus be written.

The factor in brackets, which is the ratio of the composition difference in the matrix phase to the composition difference across the phase boundary, is called the supersaturation ratio for the process, and can be determined from the tie line on the phase diagram that describes this two phase system. D is the diffusivity in the matrix phase at the composition C a , and l is the local diffusion length scale in the matrix at a surface element.

In a microstructure that is evolving by diffusion controlled growth of a b phase into an a matrix the interface velocity, v, is expected to vary with position over the ab interface. Inspection of Eq. Thus, the distribution of diffusion length scales in the structure plays a central role in determining the path and kinetics of the evolution of the microstructure. Rates of change of the global geometric properties can be related to the local physics embodied in Eq. The rate of change of the volume of the growing b phase is given by.

The kinetics of growth of the b phase is clearly determined by the temperature of the process through the diffusion coefficient in the a phase, which is strongly temperature sensitive, and the supersaturation ratio obtained from the phase diagram, which may not be as sensitive to temperature. The nucleation part of the precipitation process is implicit in the distribution of diffusion length scales.

As nucleation becomes time dependent, this distribution broadens. The time dependence of surface area and integral mean curvature can be determined by inserting the velocity Eq. Insight into the path of microstructural change for a microstructure evolving by diffusion controlled growth of the b phase can be obtained by taking ratios of these kinetic equations. The projection of the path which plots the surface area of the microstructure versus the volume of the growing phase is given by the ratio. The component of the path represented by the variation of the integral mean curvature with the volume of the growing particles is.

In both of these equations the dependence upon the diffusion coefficient in the matrix, D, and the phase diagram concentrations contained in the supersaturation ratio cancel. The path depends upon the ratio of differently weighted averages of the diffusion length scale distribution, and current values of geometric properties, S, M and W. The breadth of the diffusion length scale distribution is determined by the time dependence of the nucleation behavior in the process, i.

It is concluded that the variability of the path of microstructural evolution in diffusion controlled precipitation is largely determined by the kinetics of its topological nucleation process: nucleation. All of these results reflect the microstructology viewpoint: the description is carried as far as possible before introducing simplifying geometric assumptions.

In order to apply the kinematic equations to specific microstructural processes it is necessary to visualize the distribution of concentration gradients in the matrix at the moving interface. This evaluation is specific to each class of process. Coarsening sets in when the chemical driving force for growth of a precipitate phase has been exhausted so that the much smaller differences in composition associated with capillarity effects may begin to operate. In this case the thermodynamics of capillarity effects predicts that the concentration in the a phase at a surface element is determined by the local mean curvature of the element An element of surface on a small particle has a relatively high concentration in the adjacent a phase; surface elements on coarse particles have a relatively small concentration in the adjacent a phase.

Thus, there exists a distribution of concentrations over the collection of boundaries of the a phase. If the process is controlled by diffusion between b particles through the matrix a phase, then a concentration field evolves with time in this process. This evolution is governed by Fick's second law and the distribution of concentrations at the ab surfaces that bound the a phase.

The composition field will exhibit flux lines normal to the isocomposition contours that connect any given surface element with a communicating neighbor element. The gradient at the interface for the smaller particle can be visualized in terms of the composition difference between the surface elements which is determined by the difference in their mean curvatures and the diffusion length scale, l :.

In this result g is the surface free energy of the interface, V b is the molar volume of the b phase, R is the gas constant, T the absolute temperature H n the mean curvature of the communicating neighbor element and H that of the reference element. Insert this evaluation into the interface accommodation equation:. Note that if H is larger than H n implying the reference element is on a smaller particle v is negative and the particle shrinks. For the opposite case, the particle grows. Thus, in diffusion controlled coarsening the local velocity of an element is determined by its curvature, H, that of its communicating neighbor, H n , and a diffusion length scale, l which is of the order of the distance between particles.

Insertion of this result into the kinematic equations gives the predictions for the time evolution of global geometric properties for diffusion controlled coarsening. Insertion of this result into the remaining kinematic equations yields predictions of the rate of change of surface area and total curvature for diffusion controlled coarsening. In this case the integrations involved lead to weighted averages of the mean surface curvature distribution, as well as the distribution of diffusion length scales. These results contain no simplifying geometric assumptions. For a complete presentation of these results see reference The path of microstructural change for coarsening processes is presumed to be highly constrained.

Classical theories on the subject suggest that, after some adjustment to the initial particle size distribution, all coarsening systems of any given volume fraction approach a fixed size distribution, and thus a fixed path of microstructural evolution. Factors that might alter the path, such as the spatial distribution of the particles whether clustered or ordered have been left unexplored. The local process that governs grain growth is grain boundary migration.

In contrast with the previously described processes, grain boundary migration does not involve concentration fields, Fick's second law or diffusion length scales. The driving force for grain growth is the decrease in surface energy associated with the elimination of grain boundary area. This is accomplished if each element of boundary moves toward its local center of curvature. The classic kinetic model, due to von Neuman 27 and Mullens 28 , describes grain growth in a two dimensional grain structure. They assumed that the velocity of the boundary element is proportional to its local curvature:.

No other geometric factors are involved 4. The corresponding hypothesis for grain growth in three dimensional microstructures is that the velocity is proportional to the local mean curvature H of the grain boundary.

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## Thermodynamics Of Point Defects And Their Relation With Bulk Properties

The von Neumann result and its three dimensional analog neglect the topological processes that are an integral part of grain growth 30, The mean grain volume, which is determined by the number of grains in the structure and no other geometric parameter, can only change as a result of the topological event in which a small grain disappears, Fig.

As small grains are removed from the system new small grains are formed as a result of the topological switching processes illustrated in Fig. In the vicinity of the catastrophic topological events the surface curvatures are large and the interface velocities correspondingly high. Thus, in grain growth most interface elements are migrating slowly toward their centers of curvature but, at any instant in time, a small fraction of the surface elements that happen to be near a topological event, are moving very rapidly.

The kinematic equations may be used in principle to predict kinetics and the path of microstructural change. However, it is necessary to incorporate the observation that much of the change in surface area of the grain boundary network that occurs in any time interval is more or less directly associated with the topological events that occur in that time interval. It seems evident that this range of paths is governed by the spatial distribution of the topological events. In normal grain growth the topological events are uniformly distributed in space. In abnormal grain growth these processes apparently tend to be clustered near abnormal grains.

The material and process variables that control the path of microstructural change in grain growth remain a subject for experimental study. A geometrically general approach to modeling the sintering process requires modification of the kinematic equations. In sintering the migration of the pore solid interface is not the only contribution to the changes in the geometric properties. The annihilation of vacancies at grain boundaries plays a significant role in densification which also contributes to changes in other geometric properties. Thus integrals of the local velocity over the moving pore solid interface do not adequately describe the geometric changes that occur.

When a stack of particles is heated to a significant fraction of their melting point, necks form at points of contact. These necks are grain boundaries in the solid phase. The pore phase is a connected network which interpenetrates the solid connected network. The necks grow until they impinge, forming triple lines in the developing grain boundary network. The formation of triple lines is accompanied by the pinching off of channels in the pore network. Eventually a fully connected grain boundary network results as the pore network disconnects into isolated pores.

The thermodynamics of defect structures combines with capillarity theory to predict that vacancies are generated at the curved pore surface. These vacancies migrate to nearby grain boundaries between particles. As the vacancies annihilate at the grain boundaries, the grain centers move toward each other and the porous aggregate densifies. The process may be visualized in terms of a space filling cell structure in which each cell contains one grain and its associated porosity Each cell face is partly covered by grain boundary, partly by porosity.

Let f be the ratio of cell face area to grain boundary area at any point in the process. When the equivalent of one layer of vacancies is annihilated in the grain boundary on a particular face the volume of the cell is decreased by the removal of a layer one atom thick from the whole cell face. In the early stages of sintering, when the grain boundary occupies a small fraction of a cell face, annihilation of a layer of vacancies in the grain boundary carries with it a large volume change, i.

In the late stages of the process after porosity is isolated f approaches 1 and the volume change of the structure is roughly equal to the volume of vacancies annihilated. Sintering is an unusual microstructural process in that it involves two networks with evolutions that are coupled Assume for simplicity that the particles are single crystals. At the outset the pore network is completely connected and the grain boundary network, which is confined to the interparticle necks, is completely disconnected. As densification proceeds and pore channels begin to pinch off as growing necks impinge on each other, disconnected portions of the grain boundary structure begin to connect to form subnetworks.

At a still later stage the grain boundary subnetworks connect up to form a complete network. This grain boundary network is capable of the metric and topological changes that characterize grain growth. The subsequent path of microstructural change is strongly dependent upon the balance between grain growth and pore evolution because the elimination of porosity is most efficient for pores in the grain boundary network. The attainment of a fully dense body is favored if the pore structure and the grain boundary network remain coupled.

If the grain structure coarsens, either normally or abnormally, pores left without contact to the grain boundary network become stable and resist elimination. Paths of microstructural evolution in sintering are circumscribed primarily by the geometry of the initial powder stack embodied in the particle size, size distribution, shapes and stacking 4. Processing scenarios that involve precompaction or hot pressing versus loose stack sintering significantly alter the path in the neck growth and channel closure stages of the process but have less effect in the late stages of the process where the pores are isolated.

In that stage the coupling between the grain growth process and pore evolution determines the path. Engineering of microstructures requires wise choices based upon knowledge on the one hand of the relation between processing and the microstructural states that may be developed and, on the other hand, of the relation between microstructural states that may be achieved and their properties.

This presentation has focused on the tools that are needed to choose a material and design a process to achieve a target microstructural state. Concepts of microstructural state and the path of microstructural change form the basis for the engineering of microstructures. For a given material and initial microstructural state each class of process yields an envelope of microstructural paths. Fundamental geometric concepts must be mastered in order to describe microstructural states and paths qualitatively, then quantitatively.

A knowledge of the thermodynamics that underlie multicomponent, multiphase systems leads to an understanding of phase diagrams for real, complex systems. These maps of the domains of stability of possible equilibrium states for a system form the context within which microstructural processes occur. A set of kinematic equations relate the evolution of microstructural geometry to the distribution of velocities on boundaries in the structure. These geometrically general relationships provide a basis for establishing paths of microstructural change and rates of traverse of those paths.

These kinematic equations may be converted to kinetic equations by applying the principles of both reversible and irreversible thermodynamics to endow local interface velocities with a physical description. This connection with the physics of the moving atoms is the basis for establishing the effect that processing sequence and variables have upon the path and kinetics of microstructural evolution.

Many common microstructural processes have a topological component. The typical topological event is a local change in connectivity or number of some geometric entity in the structure. It may be accompanied by relatively large changes in local geometric properties. Rates of topological processes relative to rates of metric property changes may be responsible for introducing breadth into the spread of possible microstructural paths that a system may choose.

Engineering of microstructure presumes that a target microstructure has been circumscribed which exhibits a combination of properties desired for a particular application. The target microstructural state must be produced from a selected material through processing. It is a particular state obtained by interrupting a microstructural evolution that carries the material along some microstructural path. The spectrum of microstructural paths available from a given starting state is determined by the processing sequence and processing variables.

Understanding these connections so that the right material may be selected and subjected to the right process requires mastery of the geometry, thermodynamics, kinematics and kinetics of microstructural evolution. Rhines, F. Microstructology , Behavior of Microstructure of Materials. DeHoff, R. Topography of microstructure. Metallography , n. Quantitative Microscopy. Aigeltinger, E. Quantitative determination of topological and metric properties during sintering of copper. Minkowski, H. The quantitative estimation of mean surface curvature. AIME , n. Geometrical meaning of the integral mean curvature.

Integral mean curvature and platelet growth. Quantitative microscopy of lineal features in three dimensions. Underwood, E. Quantitative Stereology , Addison-Wesley Pub. Kurzydlowski, K. Thermodynamics in Materials Science. Hilliard, J. Specification and measurement of microstructural anisotropy.

Baddeley, A. Estimation of surface from vertical sections. Russ, J. Computer Assisted Microscopy. Schwarz, H. The characterization of the arrangement of feature centroids in planes and volumes. Gurland, J. The measurement of grain contiguity in two phase alloys. Cahn, J. The measurement of grain contiguity in opaque samples. Kattner, U. The thermodynamic modeling of multicomponent phase equilibria. Journal of Metals , p.

Fontaine, D. From Gibbsian thermodynamics to electronic structure: nonempirical studies of alloy phase equilibria, MRS Bulletin , p. Sekerka, R. Geiger, G.

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A geometrically general theory for diffusion controlled coarsening. Acta Metallurgica , n. Neumann, J. Curvature driven grain growth in two dimensions. Mullins, W. Role of grain boundary grooving in grain growth in two dimensions. Metric and topological contributions to the rate of change of boundary length in two dimensional grain growth.

Smith, C. Grain shapes and other metallurgical applications of topology.

Craig, K. Mechanisms of steady state grain growth in aluminum. A Stereological model of sintering. Science of Sintering. Services on Demand Journal. Engineering of Microstructures R. This presentation focuses upon the set of tools that must be combined to achieve this control: 1. Keywords: microstructure 1. Introduction Geometry. The scientific content of materials science is based upon a hierarchy of structures: Nuclear structure determines nuclear properties.

These tools are: 1. Geometric concepts that describe the microstructural state, quantified by the application of stereology; 2. Thermodynamics and its implementation in phase diagrams which provide the context within which processes occur; 3. Kinematics of evolving microstructures that relate local boundary motions to global geometric change; and 4. Geometry and the Microstructural State Microstructures are three-dimensional space-filling, not-regular, not-random distributions of phases and their boundaries. The geometric state of a microstructure may be reported at three levels 2 : 1.

The qualitative microstructural state; 2. The quantitative microstructural state; and 3. The topographic microstructural state. The qualitative microstructural state A microstructure is a three dimensional entity; features in the microstructure may have 3, 2, 1, or zero dimensions. The quantitative microstructural state Each of the features in the qualitative microstructural state has associated with it one or more geometric properties.

These fundamental properties fall into the following categories: 1.

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Topological properties; 2. Metric properties; a. Measures of extent; b. Curvature measures. The topographic microstructural state Concepts in the quantitative microstructural state encompass properties of individual features and the global properties of feature sets. There are three primary categories of concepts in this level of description: 1. Gradients , which report systematic variations of geometric properties with position in the structure; 2.

Anisotropies , which report systematic variations with orientation ; 3. Thermodynamics in Microstructural Design Classical phenomenological thermodynamics provides the basis for organizing information about the behavior of matter. In the following sections discussion is limited to the two major application of thermodynamics in the engineering of microstructures: 1. Thermodynamics of phase diagrams; and 2. Thermodynamics of capillarity effects.

Thermodynamics of phase diagrams The collection of equilibrium states for any system is represented as a map of the domains of stability of the various phase forms that a system may exhibit in a space that represents ranges of interest of the fundamental overarching variables, temperature, pressure and composition. Thermodynamics of capillarity effects A microstructure is a spatial distribution of geometric entities: interfaces, triple lines, quadruple points.

The conditions for equilibrium involving geometric aspects of microstructure fall into two main scenarios: 1. Kinematics and the Path of Microstructural Change In mechanics kinematics means the description of the geometry of motion as in the motion of a compound lever system without regard to the physical influences that cause the motion.

The kinematic equations There also exists a kinematics of microstructural evolution. If the boundary motions are smooth and continuous some information about what determines the path may be obtained from ratios of the kinematic equations: The sequence of geometric states, expressed in terms of how the surface area and integral mean curvature vary with the volume of the phase, is seen to be determined in a complex way by ratios of integrals over the area of the moving interface of weighted measures of the velocity distribution on the moving interface.

Topological processes in microstructural evolution Most microstructural paths result from two classes of local processes: 1. The Kinetics of Microstructural Evolution The kinematic equations describe the global geometric changes of a microstructure in terms of the distribution of interface velocities. Precipitation processes An alloy with composition C o is first solution treated at a high temperature where the phase diagram, Fig.

The interface accommodation equation for diffusion controlled growth can thus be written The factor in brackets, which is the ratio of the composition difference in the matrix phase to the composition difference across the phase boundary, is called the supersaturation ratio for the process, and can be determined from the tie line on the phase diagram that describes this two phase system.

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Understanding Electrochemical Thermodynamics through Entropy. Research Article www. The electrochemical window of Li3PS4 ranges from 0. By a series of motivated and closely linked calculations, we try to provide a portable method, by which researchers could gain insights into the physicochemical properties of solid electrolyte. Interestingly, in Liu et al. On the other hand, there are still some important physicochemical properties for Li3PS4 that remain unclear. Unfortunately, no details on the defect thermodynamics of Li3PS4 were provided.

Another important property of Li3PS4 is the phase stability during the battery operation. By a series of motivated and closely linked calculations, we try to provide a portable method by which researchers could gain insights into the physicochemical properties of solid electrolyte. The remainder of this paper is organized as follows. The computational details are outlined in Section 2.

Perfect Li3PS4 Crystal. Such frameworks stand abreast along the [] direction, keeping the opposite apex angle directions. Table 1 lists the DOI: The Li, P, and S atoms at the lattice sites are colored with pink, gray, and red, respectively. Figure 2. Table 1. This choice agrees with the previous work performed by Lepley et al.

The optimized lattice parameters agree with the experimental ones within 2. The upper and lower limits of the polycrystalline bulk and shear modulus are obtained by the Voigt and Reuss equations. Elastic Properties. Although the all-solid-state LIBs employing inorganic SEs can overcome the poor safety and low reliability of liquid electrolyte, they also have some drawbacks, and probably the most important one is the bad contact caused by the internal or external stress between the SE and the electrode.

Therefore, it is very important to get knowledge of the mechanical characteristics of SEs. Additionally, the results obtained from the atomic-scale simulations of elastic properties, such as elastic tensors and equations of state, can be used as input parameters for macroscale modeling such as the continuum medium theory, to achieve macroscale intrinsic properties of materials. The calculation details are shown in Supporting Information Section 2. Computational details on elastic properties are provided in Supporting Information Section 2. The bulk modulus measures the resistance to volume change by an applied pressure, while the shear modulus estimates the opposition to reversible shear deformations.

The shear modulus, which is a better predictor of hardness than the bulk modulus, is more dependent on C44 as mentioned above. Very recently, Ong and co-workers also reported the elastic properties of Li3PS4,44 which are consistent with our results. Because SEs could not recover these deformations, it is important to restrain deformations in the battery system. Additionally, with the increase of the E value, the covalency of a material also increases. Table 6 lists the optimized interstitial sites as well as the computed relative 7 Table 6. Compared with other anisotropic crystals reported in ref 48, Li3PS4 is moderately anisotropic.

Coordinates are given as fractional coordinates of the optimized perfect unit cell in Figures 1 and 2. The calculated defect formation, Ef i,q , and the corresponding defect concentration, S i,q , of both phases of Li3PS4 are shown in Figures 3 and 4, respectively. The vertical dotted lines in a and b represent the HSE06 band gap red and two critical voltages black : 0.

Li3PS4 would become unstable when the voltage is above 3. The stability of Li3PS4 facing inert electrodes is studied, adopting the method put forward by Ong et al. The dotted lines in a and b represent the HSE06 band gap red and two critical voltages black : 0.

## Download Thermodynamics Of Point Defects And Their Relation With Bulk Properties

In this voltage range, these two point defects have an equal concentration about 3. Above 3. The equilibrium phases with Li3PS4 are noted in black stable and red unstable , respectively. The insert in g shows the enlarged convex hull near Li3PS4. We list the obtained phase equilibria of Li3PS4 in equilibrium with explore this interfacial passivation, we examine the phase equilibria of Li3PS4 when it faces metallic lithium.

Employing the approach proposed by Ong et al.