Transport mechanisms and densification during sintering: II. Grain boundaries

Transport mechanisms and densification during sintering: II. Grain boundaries

Chemical Engineering Science 64 (2009) 3810 -- 3816 Contents lists available at ScienceDirect Chemical Engineering Science journal homepage: w w w ...

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Chemical Engineering Science 64 (2009) 3810 -- 3816

Contents lists available at ScienceDirect

Chemical Engineering Science journal homepage: w w w . e l s e v i e r . c o m / l o c a t e / c e s

Transport mechanisms and densification during sintering: II. Grain boundaries Hadrian Djohari, Jeffrey J. Derby ∗ Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455-0132, USA

A R T I C L E

I N F O

Article history: Received 8 January 2009 Received in revised form 2 March 2009 Accepted 13 May 2009 Available online 21 May 2009 Keywords: Sintering Transport processes Materials processing Mathematical modeling Microstructure Powder technology

A B S T R A C T

Finite element, meso-scale models provide a means to probe the mechanistic driving forces for particle evolution during sintering and were applied in a companion paper [Djohari, H., Martínez-Herrera J., Derby, J.J., 2009. Transport mechanisms and densification during sintering: I. Viscous flow versus vacancy diffusion. Chem. Eng. Sci., in press, doi:10.1016/j.ces.2009.05.018.] to compare different behaviors of the sintering of glassy particles by viscous flow and the sintering of idealized crystalline systems without a grain boundary via vacancy diffusion. Here, the effects of a grain boundary are included in the meso-scale model and resultant behavior is compared to prior cases. A grain boundary acts as a sink for vacancies, drawing a flux toward itself and allowing for their accumulation and collapse. The resultant solid-body motion of the particles leads to significant shrinkage at the onset of sintering; neck growth with little shrinkage was observed in systems without a grain boundary. These effects are scaled by the magnitude of the grain boundary diffusivity and the size of the dihedral angle. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction Understanding the physical mechanisms that drive the sintering of ceramic and powder metallurgical solid-state materials is important for controlling the resultant properties of the material. In a companion paper (Djohari et al., 2009), finite element models were applied to solve for coupled materials transport and morphological evolution during the sintering of two initially spherical particles by viscous flow and by vacancy diffusion. These analyses showed how different materials transport mechanisms fundamentally changed the character of the geometrical evolution responsible for neck growth and densification. In short, the long-range nature of viscous flows acting to sinter glassy particles resulted in simultaneous neck growth and shrinkage at all times, while the localized nature of vacancy diffusion during the sintering of crystalline particles promoted neck growth early on but without significant densification until much longer times. A curious result occurring when the surface diffusion of vacancies is dominant is an elongation of the two particles before any shrinkage of the system occurs. However, the models for crystalline sintering employed in Djohari et al. (2009) are highly idealized, since they do not consider the effects of a grain boundary between the particles. In this paper, we present a grain boundary model appropriate to describe the sintering

∗ Corresponding author. Fax: +1 612 626 7246. E-mail address: [email protected] (J.J. Derby). 0009-2509/$ - see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2009.05.022

of two particles and describe the new character of materials transport and its effect on densification. While the two-particle system considered here is overly simple compared to the geometry of real powder compacts, the mathematical model is rigorous and the resultant insight to materials transport and morphological evolution is meaningful. A more complete overview on sintering models is provided in Djohari et al. (2009), so here we concentrate instead on prior particlescale (sometimes referred to as meso-scale) models for the sintering of crystalline materials. Nichols and Mullins (1965) and Nichols (1966, 1968) were the first to solve for the self-consistent evolution of surface geometry coupled with surface diffusion processes in a ¨ two-dimensional geometry. Easterling and Tholén (1970) solved a similar problem for volume diffusion processes. German and Lathrop (1978) simulated the sintering of spherical powder particles via surface diffusion. Bross and Exner (1979) were among the first to consider the mechanism of grain boundary diffusion coupled with volume diffusion. Carter and Cannon (1988) and Cannon and Carter (1989) analyzed the equilibrium shapes of rows of sintered particles for various dihedral angles associated with grain boundaries. Svoboda and Riedel (1995a,b) consider a more complicated, twodimensional configurations of six cylinders sintering with surface and grain boundary diffusion. In the first of several significant advances to account for grain boundary effects using a continuum modeling approach, Pan and Cocks (1995) and Pan et al. (1997) developed a variational formulation to model grain boundary and surface diffusion which was solved via finite element and finite difference methods. Sun et al. (1996) and

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Sun and Suo (1997) developed similar approaches to model grain motion and surface diffusion but applied their models to thin films. Pan et al. (1998) continued development to model microstructure evolution by a combination of grain boundary diffusion, grain boundary migration, and surface diffusion. Parhami et al. (1999) followed with an analysis of the sintering of a line of spherical crystalline particles. Zhang and Schneibel (1995), Zhang and Gladwell (1998), and Zhang et al. (2002) also produced notable advances to address the formulation and solution of sintering and microstructural evolution problems involving coupled surface diffusion and grain boundary motion. Ch'ng and Pan (2004, 2005) have developed improved means of representing a larger number of grain boundaries and pores to model the evolution of microstructure in two-dimensional, planar solid. Wakai and Aldinger (2006) have applied the phase-field approach to model concurrent materials transport mechanisms and grain boundary effects in the sintering of multiple, two-dimensional particles. Wang (2003), Wakai (2006) and Wakai et al. (2007) have modeled several curvature-driven transport processes to address more complicated, three-dimensional arrangements of particles during sintering. In the ensuing discussion, we briefly review the model development for coupled vacancy diffusion and particle surface motion during the sintering of two, spherical crystalline particles with a grain boundary. Our primary emphasis is not on the development of new modeling capabilities. Rather, following the discussion in Djohari et al. (2009), we desire to discuss the profound impact that a grain boundary has on densification behavior via a mechanistic understanding of the underlying materials transport processes. 2. Model formulation 2.1. Governing equations The presence of a grain boundary introduces a significant complexity to the system, as evidenced by extensive development efforts in prior sintering models for crystalline systems; see, e.g. Zhang and Schneibel (1995), Zhang and Gladwell (1998), Zhang et al. (2002), Pan and Cocks (1995), Pan et al. (1997, 1998), and Ch'ng and Pan (2004, 2005). A grain boundary exists when two particles of different crystalline orientation are joined. It is assumed that this grain boundary is of finite width and, due to its atomic structure, vacancies can readily move along it, as described by a grain boundary diffusion coefficient, Dgb . The accumulation of vacancies along a grain boundary leads to a phenomenon often referred to as plating, where, in essence, vacancies align to form a disk of open volume, which collapses by the attraction of the adjacent grains and leads to a net inward motion of the particles. This solid-body motion of the particles results in shrinkage or densification. In the present study, we consider only the simplest case of the sintering of two identical spheres, so the grain boundary separating these particles can be assumed to be straight and immobile throughout the sintering process. We also assume that the crystalline particles are isotropic and that no external stresses are applied to the system. Real crystalline particles are anisotropic; however, the assumption of isotropic transport properties for vacancies will not substantially impact the general understanding presented by this model. In addition, under conditions of no external stress, Nabarro–Herring or Coble creep associated with vacancy diffusion will not be active. Here, vacancy diffusion is driven solely by effects arising from the initial curvature of the particle surfaces. For a mathematical description, we fix a cylindrical coordinate system at the exact center of the two particles, and consider the transport of vacancies throughout the system via volume diffusion and surface diffusion; see the schematic diagram in Fig. 1. In this coordinate frame, the grain boundary is fixed and the continuous collapse of vacancies along the grain boundary results in solid-body

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Fig. 1. Schematic diagram of the two-particle sintering model considered here.

motion of the particles toward the grain boundary, as indicated in Fig. 1. We represent the plating velocity by computing the rate of accumulation of vacancies along the grain boundary and performing a mass balance as the vacancies collapse. This is achieved by balancing the flux of vacancies diffusing along the particle surface toward the grain boundary triple junction (this flux is driven by the gradient of chemical potential as a function of mean surface curvature) with the flux of vacancies flowing along the grain boundary (which is driven by gradients in chemical potential as a function of stress along the grain boundary). For brevity, we do not perform a detailed derivation here, rather the interested reader is referred to Zhang and Schneibel (1995) and Zhang and Gladwell (1998). The outcome of this analysis leads to a nondimensional plating velocity, vgb = v˜ gb a2 k/Ds , with the tilde representing the dimensional plating velocity, a representing the initial particle radius, k as Boltzmann's constant,  as absolute temperature, Ds as the surface diffusivity of vacancies,  as surface energy, and  as the volume occupied by a vacancy. The resultant velocity of the particles is then given by vgb =

4b x3



 2 sin

 2



 − T x (−ez ),

(1)

where vgb denotes the solid-body velocity vector for particle motion,

b = Dgb /Ds is the ratio of grain boundary and surface diffusivities,

x = x˜ (t)/a is the dimensionless triple junction position (which is simply the radius of the neck),  is the dihedral angle, T is the dimensionless local curvature of the particle surface at the triple junction, and ez indicates the unit vector pointing in the z-coordinate direction (see Fig. 1). The plating motion is thus directed inward, toward the grain boundary, resulting in shrinkage of the system. It is important to emphasize that this plating motion is not driven by external forces, rather it arises from the underlying accumulation and annihilation of vacancies along the grain boundary. There is no creep of the material in the particle, rather there is a simple solid-body motion (a collapse) of the entire particle adjacent to the grain boundary with a velocity of vgb . We refer the reader to Zhang and Schneibel (1995), Zhang and Gladwell (1998), and Djohari (2004) for a more detailed derivation of this condition. In our prior models for vacancy diffusion Djohari et al. (2009), the particle material remains motionless with respect to the coordinate system, while the material within the particles rearranges during sintering. However, in the presence of a grain boundary, the plating motion of the particles creates an additional convective transport mechanism for vacancies. Namely, vacancies can also be transported through the system by motion of the crystalline material containing them. The importance of this transport mechanism is estimated by

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a Peclet number for vacancy transport, Pe =

av˜ gb Dv

,

(2)

which is a characteristic ratio of the flux of vacancies transported by convection to that by diffusion. To estimate the magnitude of this quantity, we note that a characteristic value for the initial particle radius is a ≈ 10−3 –10−4 cm, the grain boundary plating velocity is typically v˜ gb ≈ 10−8 –10−7 cm/s, and typical values for vacancy diffusivity are Dv ≈ 10−10 –10−14 cm2 /s. Therefore, we estimate Pe ≈ 10−2 –104 , which is significant (order unity or larger) in many cases. Interestingly, neither the prior derivations of Zhang and Schneibel (1995), Zhang and Gladwell (1998), and Zhang et al. (2002) nor Pan and Cocks (1995), Pan et al. (1997, 1998), and Ch'ng and Pan (2004, 2005) seem to account for this vacancy transport mechanism; however, Wakai and Aldinger (2006) does include it in his formulation. Accordingly, we modify the transport equations presented previously (Djohari et al., 2009) so that the balance for vacancies within the particles is (written in dimensionless form) ∇ 2 c − vgb · ∇c = 0,

(3)

where ∇ represents the gradient operator in the coordinate system defined for this problem and the dimensionless vacancy concentration, c is defined as c ≡ c˜ /c0 , where c0 is the equilibrium vacancy concentration along a flat surface. In this choice of nondimensionalization, the length is scaled with the initial particle radius a and time is scaled as t ≡ t˜Dv c0 /a2 , where the tilde represents the dimensional time, and Dv is the volume diffusion coefficient for vacancies. Surface stress is related to vacancy concentration along the surface via the Gibbs–Thomson equation, written in dimensionless form as cx (1 − c) = ,

(4)

where  is the dimensionless mean curvature of the surface and cx = ka/  is a materials constant. This condition prescribes the surface vacancy concentration, which is used to supply a boundary condition to the governing equation (3) for vacancies in the bulk. To solve Eq. (4) requires specification of the slope of the particle surface at the symmetry plane between the two particles, which sets the dihedral angle,  (see Fig. 1). In our idealized analyses of the companion to this paper, we assumed that the dihedral angle was 180◦ in the absence of a grain boundary (Djohari et al., 2009). Here, however, we specify a slope of the particle surface at the triple junction so that a specific dihedral angle is achieved. That this dihedral angle exists as a materials property is a result of the different interfacial energies of the particle surface and the grain boundary and the requirement that their junction be at equilibrium. We consider several values of  in the ensuing computations to investigate a range of different materials and behaviors. Finally, we focus on a balance of material at the surface of the particles. Estimating a vacancy Peclet number based on surface diffusion also yields estimates of Pes = av˜ gb /Ds > 1 and argues for including a convective transport term in the governing equation for surface transport. We thus write the following nondimensional equation to describe the surface, n · x˙ s + n · vgb = n · ∇c + ∇s2 c + vgb · ∇s cˆ

at xˆ = xˆ s .

(5)

Here, n is the unit vector pointing in the normal direction of the particle surface, xs is the surface position in the fixed coordinate system, and x˙ s is the time derivative of the surface position. The total normal motion of the interface is represented by the terms on the left-hand side of the above equation; notice especially the second term that accounts for how the solid-body motion of the particle affects the position of the interface. On the right-hand side of

Eq. (5), terms (from left to right) account for the flux of material to the surface by the diffusion of vacancies through the bulk, the net flux of material to a point on the particle via the surface diffusion of vacancies, and the convective transport of vacancies along the surface by the plating velocity of the particles. The parameter  = Ds /Dv c0 represents a ratio of time scales for surface and volume diffusion, and ∇s represents a dimensionless surface gradient operator, which is written in terms of arc length along the particle surface. At the triple junction, there must be a balance of vacancy fluxes between the particle surfaces and the grain boundary. Consistent with the flux of vacancies at the grain boundary driving the particle plating relationship, Eq. (1), the following boundary condition is derived and applied for the solution of the surface conservation equation, Eq. (5),    2 (6) − T x . −t · ∇s c = 2b 2 sin 2 x Here, t corresponds to a unit vector tangent to the particle surface at the triple junction. An initial particle geometry completes the specification of this time-dependent problem. More details on the complete derivation are available in Djohari (2004). 3. Numerical methods The solution of the governing equations posed above requires a robust and accurate numerical method, owing to the movingboundary nature of the problem, where the geometry of the system evolves in time and therefore must be computed in concert with the underlying vacancy transport problem. In addition, the coupled solution of surface diffusion and interface motion is especially challenging, since the accurate computation of surface curvature is required. This challenge alone has limited many prior models to computing relatively simple geometries accompanied by relatively small-scale surface motion. We implement a finite element method with front-tracking to solve for material transport and morphological evolution, similar to approaches that have been effectively employed for other movingboundary problems (Ruschak Kenneth, 1980; Ungar et al., 1988; Sackinger et al., 1996; Salinger et al., 1994; Hooper et al., 2001). Accuracy is achieved by the use of second-order polynomial basis functions defined over a moving mesh with elements graded in size to capture sharp gradients in space and via second order, implicit temporal integration techniques for computing the evolution of the system. We refer the reader to the companion paper (Djohari et al., 2009) for a slightly expanded presentation of our numerical implementations and to Martínez-Herrera (1995) and Djohari (2004) for more detailed discussions. In Martínez-Herrera (1995) and Djohari (2004), we also present several computations to verify our codes, including comparison with several analytical mathematical solutions derived for idealized cases representing viscous sintering (Hopper, 1984) and coupled surface and grain boundary diffusion for various dihedral angles (Cannon and Carter, 1989). 4. Results Our primary goals are to understand the transport mechanisms that drive sintering and the resultant morphological evolution that defines the sintered state, such as the degree of neck growth and densification. In the following discussion, we examine the behavior of this system primarily with respect to the surface-to-volume diffusivity ratio, , the grain boundary-to-surface diffusivity ratio, b , and the dihedral angle, . Each of these parameters is defined primarily by the material properties of the system under consideration. However, each can also be modified to a certain extent by process

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Fig. 3. The final, equilibrium shape after the sintering of two, initially spherical, crystalline particles of equal size is shown for various dihedral angles, .

Fig. 2. Geometrical evolution and instantaneous surface vacancy concentration levels are shown for the case of two, initially spherical, crystalline particles of equal size that sinter by combined surface, volume, and grain boundary diffusion (for  = 1, b = 0.1, and  = 120◦ ). Dimensionless times for each case are: (a) t = 0; (b) t = 0.1; (c) t = 1; (d) t = 5; (e) t = 20; and (f) t = 100.

changes. For example, since diffusivities typically are controlled by different activation energies, their relative ratios can be influenced by system temperature. The addition of different constituents into the system can alter surface and grain boundary energies, thus modifying the dihedral angle. Finally, while we do not consider such effects here, the sintering trajectory is also affected by the initial particle geometry (here taken to be equally sized spheres with a given neck size), so that, for example, the sintered state can be affected by changes in particle size distribution and the application of stress before sintering to alter the initial geometries of the necks between particles. 4.1. Particle geometry and transport mechanisms Fig. 2 shows the results of a computation for the sintering of two initially spherical crystalline particles for parameters which are estimated to approximate those of a typical ceramic material such as alumina (Johnson and Cutler, 1963; Gontier-Moya et al., 1998), namely a surface-to-volume diffusivity ratio of  = 1, a grain boundary-tosurface diffusivity ratio of b = 0.1, and a dihedral angle of  = 120◦ . Under these conditions, volume diffusion of vacancies through the particles is relatively unimportant, and surface diffusion serves as the primary vacancy transport mechanism Djohari (2004). Threedimensional representations of the shape of the system at various times are shown along with the instantaneous concentration of vacancies on the surface of the particles indicated by color (with blue representing low and red representing high vacancy concentrations). Note that the scalloped shapes are an artifact of a rather coarse azimuthal discretization used by the visualization scheme; this is not indicative of the much finer mesh that was used for the computations.

The initial configuration at dimensionless time t = 0 shows two spherical particles just touching to form a neck. After a short period of time, shown in Fig. 2(b) for t = 0.1, the neck radius has increased significantly and the particles have moved closer together. The presence of the grain boundary results in a remarkable change in behavior compared to the crystalline cases where it has been omitted in Djohari et al. (2009), as indicated by the vacancy distribution along the particle surface. Without a grain boundary, the concentration of vacancies is always highest at the thinnest point of the neck joining the particles and vacancies always diffuse away from the neck, leading to a net flow of material toward the neck. The surface colors of the case in Fig. 2(b) indicate that vacancies will still diffuse from the red (high vacancy concentration) band outward toward the blue (low concentration) ends of the particles as in the cases without the grain boundary, leading to a net flux of material from the outer lobes to the neck. However, there is a fundamental change in the nature of vacancy transport adjacent to the grain boundary. Namely, the grain boundary acts as a strong sink for vacancies, resulting in a ring of very low vacancy concentration along the surface where the particles meet; see the blue band at the particle junction in Fig. 2(b). With a grain boundary, vacancies diffuse toward and into the grain boundary from nearby surface areas; this is the reverse of the behavior of a crystalline system without a grain boundary Djohari et al. (2009). Vacancies flowing into the grain boundary are then balanced by their rate of collapse along the grain boundary and lead directly to the solid-body, inward motion of the particles themselves, also known as the plating velocity of the particles. As sintering proceeds, this two-way vacancy flux persists, with minima in the vacancy concentration at the grain boundary and the ends of the particles and with the highest surface vacancy concentration slowly shifting outward along the surface with time; see Figs. 2(c)–(e). Fluxes of vacancies toward the ends of the particle continue primarily to grow the neck, while fluxes of vacancies into the grain boundary act to densify (shorten) the system. Fig. 2(f) shows the system after a very long time. Here, the vacancy concentration is uniform over the surface, neck growth, and shrinkage have stopped, and the system has reached an equilibrium. Again, unlike the cases with no grain boundaries (and therefore a dihedral angle of 180◦ ) considered before Djohari et al. (2009), the presence of the grain boundary strongly affects the equilibrium shape that is approached by the system after long times. Fig. 3 shows the long-time shapes of the system for several dihedral angles, .

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Fig. 4. Dimensionless shrinkage for the sintering of two initially spherical particles is plotted as a function of the dimensionless neck radius to indicate densification and neck growth. Note that time is implicit in this representation and increases from lower left to upper right. Each system evolves according to a different characteristic time scale.

Each of these shapes represents a system surface that simultaneously satisfies the dihedral angle at the grain boundary, minimizes the surface area (and thus the surface energy) of the system, and conserves the volume (mass) of the initial configuration of the two spherical particles. As is readily seen, increasing the dihedral angle allows for a more complete approach of the particle centers and a greater degree of shrinkage and neck growth. 4.2. Neck growth versus particle shrinkage The morphological evolution of this simple system can be related to structural changes expected to take place in a more complicated powder compact. In a two-particle system, the most important features of morphology are the dimensionless neck radius, x/a, and the change in length of the system, which is directly related to shrinkage. We employ a dimensionless shrinkage defined as the change in the length of the system, l ≡ l0 − l(t), where l0 represents the initial length and l(t) is the total length at time t, scaled by the initial distance separating the particle centers, 2a. We comment that the following results are discussed in terms of geometrical evolution, rather than dimensionless time, in order to permit comparisons between different systems which may exhibit widely varying time scales for sintering. Fig. 4 plots the dimensionless shrinkage as a function of neck radius for the prior cases of viscous sintering, volume diffusion sintering, and surface diffusion sintering (both without grain boundaries) from Djohari et al. (2009) along with that for a case featuring simultaneous vacancy diffusion and grain boundary mechanisms (with  = 1 and b = 0.1). For a consistent comparison, a dihedral angle of  = 180◦ is used for the grain boundary computations plotted here, which is the same angle as that employed for the prior systems. As discussed in Djohari et al. (2009), the case of viscous flow shows concurrent neck growth and shrinkage, while the crystalline systems without grain boundaries feature neck growth with nearly no densification until later stages. The case of vacancy diffusion with a grain boundary considered here shows a behavior which lies between the viscous case and the no-grain boundary crystalline cases. Namely, the grain boundary case allows for significant shrinkage early on, consistent with the conventional wisdom that grain boundaries are

Fig. 5. The effect of the grain boundary to surface diffusivity ratio, b , is shown for the case of the sintering of two initially spherical particles via surface diffusion ( = 0) with a dihedral angle of  = 180◦ ; geometrical evolution is shown via a plot of dimensionless shrinkage versus dimensionless neck radius. Note that time is implicit in this representation and increases from lower left to upper right. Each system evolves according to a different characteristic time scale.

necessary for densification of crystalline systems (Easterling and ¨ Tholén, 1970; Rahaman, 1995). There is an inflection in the curve of the grain boundary case occurring at approximately x/a = 1, where shrinkage slows until rapidly increasing near the final stages of relaxation to a single sphere at equilibrium. This corresponds to a transition occurring after a significant amount of time between a sintering regime where grain boundary effects are dominant to one where the geometrical evolution of the surface proceeds mostly by surface diffusion. The effect of the magnitude of grain boundary diffusion on morphological evolution is shown in Fig. 5, where dimensionless shrinkage is plotted as a function of neck radius for cases with no volume diffusion ( = 0), a dihedral angle of  = 180◦ , and the grain boundary-to-surface diffusion parameter, b taking on several values. As the importance of grain boundary diffusion increases with increased values of b , the amount of shrinkage increases, showing the role of the plating action at the grain boundary. Interestingly, the elongation behavior of the system (discussed extensively in Djohari et al., 2009 for the case of pure surface diffusion), is mitigated as the rate of grain boundary diffusion increases. At higher values of b , the two-particle system consistently shrinks as the neck grows; however, at lower grain boundary-to-surface diffusion ratios, such as the b = 0.001 case shown here, there is an initial period of shrinkage until x/a ≈ 0.75, followed by a period of elongation until x/a ≈ 0.9, and a final shrinkage to the equilibrium state. Fig. 6 shows shrinkage as a function of neck radius for cases with grain boundary and vacancy diffusion ( = 1 and b = 0.1) as the dihedral angle  is varied. An immediate observation on these results is that the total amount of system shrinkage and the final neck radius decreases as the dihedral angle decreases, as indicated by the termination points for each curve. This behavior is consistent with the behavior shown in Fig. 3 for the final, equilibrium states reached by the system. Interestingly, the rate of densification with neck growth increases, but only slightly, as the dihedral angle is decreased. Apparent in all cases is a break in the slope of the curves at later times, indicating a transition from grain boundary dominated behavior to surface and volume diffusion effects. This behavior was also seen by Zhang and Schneibel (1995) in their analyses.

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Fig. 6. Dimensionless shrinkage is plotted as a function of the dimensionless neck radius for the sintering of two initially spherical particles via surface, volume, and grain boundary vacancy diffusion ( = 1 and b = 0.1) for several values of the dihedral angle . Note that time is implicit in this representation and increases from lower left to upper right. Each system evolves according to a different characteristic time scale.

5. Summary and discussion We have described a meso-scale, finite element model for the sintering of two identical, initially spherical crystalline particles separated by a grain boundary. Following the derivation of Zhang and Schneibel (1995) and Zhang and Gladwell (1998), vacancy diffusion and accumulation along the grain boundary are described along with the resultant solid-body particle motion driven by the collapse of vacancies at the grain boundary. We also argue that this plating motion need be accounted for by convective terms in the evolution equations for vacancy transport and surface motion, thereby extending the derivations of prior models (Zhang and Schneibel, 1995; Zhang and Gladwell, 1998; Zhang et al., 2002; Pan and Cocks, 1995; Pan et al., 1997, 1998; Ch'ng and Pan, 2004, 2005). As argued in our prior paper (Djohari et al., 2009), such a model is a rigorous representation of an admittedly simplified geometry for the modeling of a real powder compact. Nevertheless, such an approach provides a fundamental understanding of the role of transport mechanisms on setting the morphological evolution of sintering particles, thus yielding insight to more general sintering behavior. The presence of a grain boundary between crystalline particles that are sintering by vacancy diffusion mechanisms leads to profound effects. Without a grain boundary, vacancy concentration is highest at the neck and vacancies diffuse outward. In contrast, there is always a two-way flux of vacancies when a grain boundary is present; vacancies diffuse both inward to the grain boundary and outward toward the particle ends. In addition, the energetic effects of the grain boundary establishes a dihedral angle that must be satisfied at all times, thus limiting the extent to which the two particles can sinter together. The diffusion of vacancies to the grain boundary, followed by their accumulation and collapse, drives a plating motion that significantly affects the rate of shrinkage of the system. Indeed, our computations show that the shrinkage as a function of neck radius for crystalline particles with a grain boundary approaches that of glassy particles undergoing viscous flow. Unlike the crystalline cases of volume or surface diffusion with no grain boundary, where almost no shrinkage occurs at early stages of neck growth, the solid-body particle motion driven by the grain boundary causes substantial shrinkage from the onset of sintering. These results affirm the conventional sintering

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wisdom that, while neck growth may occur by surface or bulk diffusion of vacancies, crystalline systems rely on grain boundaries to ¨ promote densification (Easterling and Tholén, 1970; Rahaman, 1995). Our results also show an interesting interplay between grain boundary and surface diffusion with respect to geometrical evolution. Namely, the elongational effects observed in Djohari et al. (2009) for the case of pure surface diffusion (with no grain boundary) can also affect the shrinkage of the system with a grain boundary. Higher grain boundary diffusion rates override the elongation at later stages and cause the system to always contract, albeit at different rates. However, for lower values of the grain boundary diffusivity, namely for values of b ⱕ 0.001, the two-particle system starts by shrinking, then expands, and finally contracts again. The occurrence of such behavior in a real powder compact would result in a very complicated stress history through space and time for the sintered piece. Finally, our results also demonstrate that the densification attained by the sintering of crystalline systems with grain boundaries will be affected by the value of the dihedral angle formed between particles. Our transient computations indicate that the primary effect of dihedral angle on geometrical evolution is that smaller angles increase the shrinkage rate early on but eventually limit the extent of densification at later stages. This late-stage behavior is consistent with the prior analyses by Cannon and Carter (1989). All in all, the most significant effect of grain boundaries demonstrated by the computations shown here is the solid-body motion of the particles induced by vacancy accumulation and collapse. In the very simple geometry considered here, the plating motion is directed solely toward the centerline between the two particles, producing a significantly increased rate of shrinkage. However, the effects of this motion will be much more complicated in a powder compact, where multiple particle contacts will form during sintering. Different plating rates in multiple directions will give rise to particle and grain boundary movement and rotations and a complex stress distribution. Representing such complexity is the challenge for continued development of meso-scale mathematical models for sintering. The next advances will require more realistic models of complicated powder compacts, especially three-dimensional geometries (Zhou and Derby, 1998, 2001; Wakai, 2006) and the representation of many particles (Zhang et al., 2002; Ch'ng and Pan, 2005; Wakai and Aldinger, 2006; Wakai et al., 2007). Meaningful advances will also be accomplished by linking these fundamental models of the meso-scale to larger-scale continuum mechanical models (Kraft and Ridel, 2004; Olevsky et al., 2006; Kuzmov et al., 2008). Acknowledgments Acknowledgment is made to the Donors of the American Chemical Society Petroleum Research Fund and to the Minnesota Supercomputing Institute for partial support of this research.

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