Chap.3 solid-state sintering fundamentals 2007.4.9 Eun Ju PARK
Contents Basic concepts Assumptions of sintering theories Coarsening and densification Viscous flow Evaporation-condensation Surface diffusion Volume diffusion Grain boundary diffusion Plastic flow Overview of mass transport processes Adhesion, rearrangement, and repacking Initial-stage neck growth Intermediate stage Final stage Coarsening versus densification Theoretical introduction Sintering stress Stages of sintering Data analysis Calculations of sintering rates Sintering diagrams Summary Mass transport mechanisms
Basic Concepts Sintering forms solid bonds between particles when they are heated. ∆ ( γ A) = ∆ γ ∙ A + γ ∙ ∆ A ∆ γ ∙ A Densification γ ∙ ∆A Coarsening ∆ ( γ A) Densification and Coarsening
The temperature needed to induce sinter bonding versus densification depends on the material and particle size . Most materials exhibit sintering at homologous sintering temperatures between 0.5 and0.8. With higher temperatures, longer times, or smaller particles, the bond grows more rapidly and densification becomes evident.
Adhesion Initial Intermediate Final This stage occurs when particles come into contact. A weak cohesive bond Rapid growth of the interparticle neck. The pore structure becomes smooth and develops an interconnected. Giving a larger average grain size with fewer grains. The pores are spherical and closed, grain growth is evident.
Assumptions of Sintering Theories <ul><li>monosized spheres in point contact </li></ul><ul><li>sinter under isothermal condition </li></ul>In reality, most sintering is performed using nonspherical powders, with wide particle size distributions and compaction prior to sintering. Compaction repacks the particles, collapses large pores, and enlarges the particle contact but may introduce new defects. Much of the bonding between particles occurs prior to attaining the maximum temperature.
Coarsening and Densification Sintering of Boron ( coarsening with out densification ) It is common to observe sintering with simultaneous densification and coarsening.
Sintering stress The stress from interfacial energies acting over curved surfaces in a sintering system is termed the sintering stress. : The Laplace equation For a small neck, the gradient can be quite large. Thus, the stress gradient provides a driving force for mass flow to the neck. As the neck grows, the curvature gradient is relaxed and the proess slows.
MASS TRANSPORT MECHANISMS Surface transport (no shrinkage or densification) Evaporation-Condensation Surface diffusion Volume diffusion Bulk transport (shrinkage or densification) Plastic flow Grain boundary diffusion Volume diffusion Viscous flow low-temperature Low-stability Only for compacted powders, where the initial dislocation density is high. Most crystalline materials Amorphous materials
Viscous Flow Over a limited temperature range, an amorphous material has a viscosity that varies with a temperature dependence . Q=activation energy η 0 =proportionality coefficient T=absolute temperature k=boltzmann’s constant Early during isothermal sintering the neck diameter X between particles of diameter D grows in proportion to the square root of the sintering time. γ =surface energy t=time Since higher temperatures lower the viscosity, there is a progressive increase in neck size with temperature.
Sintering data for glass spheres, where the square Amorphous materials is nonexistent a grain boundary at the sinter bond. Consequently, as neck growth proceeds, amorphous materials can readily achieve a zero curvature condition where the convex and concave radii are equal but opposite in sign.
Evaporation-Condensation Vapor transport during sintering leads to the repositioning of atoms located on the particle surface, without densification . P=equilibrium vapor pressure T=temperature P 0 =material constant Q=activation energy for evaporation k=boltzmann’s constant Higher temperatures give a higher vapor-pressure and more vapor-phase transport, since the flux depends on the evaporation rate. Consequently, evaporation occurs preferentially from flat or convex particle surfaces. Preferential deposition occurs at concave necks between particles where the vapor pressure is slightly below equilibrium. For many materials evaporation-condensation transport is slow at typical sintering temperatures. Arrhenius’s equation
<ul><li>small powders with high susrface areas and high vapor pressure </li></ul><ul><li>sintering atmosphere (addition of an active chemical species) </li></ul>
Once the neck size reaches an equilibrium dictated by the solid-vapor dihedral angle, further neck growth depends on grain growth. In final-stage sintering, closed pores become distorted by pore migration with the moving grain boundaries.
Surface Diffusion 1 step Breaking an atom away from existing bonds, at a kink depends on both the surface orientation and temperature 2 step Atom moves with a random motion across the surface usually a fast step 3 step The atom must reattach at an available surface site, possibly again at a kink <ul><li>Surface diffusion initiates at lower temperatures in comparison with other </li></ul><ul><li>sintering mechanisms. </li></ul><ul><li>No shrinkage </li></ul>terrace ledge
Volume Diffusion Volume diffusion adhesion From the neck surface through the particle interior, with subsequent emergence at the particle surface. (no densification or shrinkage) Volume diffusion densification It involves vacancy flow to the interparticle grain boundary from the neck surface. Dislocations and vacancies Vacancies can be emitted or annihilated by dislocations via a process termed dislocation climb.
Volume diffusion -concetration of vacancies Concentration of vacancies -> temperature , curvature effect Mass flows into the neck region due to vacancy concentration differences in the sintering microstructure. C o =equilibrium vacancy concentration γ =surface energy Ω =atomic volume k=boltzmann’s constant The more highly curved the surface, the smaller R1 and R2 and the greater the departure from equilibrium. -> leads to mass flow into the neck Sintering rate : Fick’s first law J=flux in terms of atoms or vacancies per unit area per unit time D V =the diffusivity <ul><li>Compound -> temperatrer, stoichiometry </li></ul><ul><li>Off-stoichiometric ionic compounds contain </li></ul><ul><li>excess vacancies to neutralize charge. </li></ul><ul><li>elimination of the small pore </li></ul>
Grain Boundary Diffusion Grain boundary diffusion is relatively important to the sintering densification of most metals and many compounds. A sketch that visualizes the repeated defect Structure associated with a grain boundary Having a 36.9° misorientation between grains. As sintering progresses, transport takes place between pores via the grain boundary, leading to pore coarsening. Vacancy accumulation on a grain boundary requires motion of the boundary, and this is resisted by contacting neighbors. It is a high grain boundary energy that is a prime cause of simultaneous grain growth during sintering. -> segregates to grain boundary (to add a species, Ni,Fe, W, Mo,Fe,Cu…)
Plastic Flow Plastic flow is the motion of dislocations under stress. The dislocation flow is restricted to the early stage of sintering. : As the neck enlarges, the shear stress declines and falls below the flow stress for the material and the process becomes inactive. Siegel Dislocation participate in sintering during heating, especially if the powders was subjected to plastic deformation during compaction. Schatt and co-workers Demonstrated densification rate improvements because of dislocation climb with the rate of pore elimination.
Adhesion, Rearrangement, and Repacking <ul><li>Adherence occurs due to weak forces, including van der Waals forces and agglomeration </li></ul><ul><li>forces from liquids. </li></ul><ul><li>The particles will ratate and repack to obtain a higher packing density and lower-energy </li></ul><ul><li>grain boundary structure. </li></ul>
Initial-Stage Neck Growth The initial stage ends when the necks begin to impinge at approximately a neck size ratio X/D of 0.3. Assume that sintering occurs between two equal spheres with conservation of volume by a single mass transport mechanism. Here is a gradient in the curvature over the sintering geometry. The curvature gradient drives the mass flow, by any of the mechanisms discussed above, to smooth the surface and possibly densify the structure. Consider surface-transport-controlled sintering where at any point on the surface curve, The instantaneous change in the neck profile, At the point v on the neck profile the principal radii of curvature The local curvature κ ,
The flux depends on the curvature gradient at each point and the mobility of atoms, meaning that the neck volume change depends on the arrival rate for mass at the sinter bond. J=the atomic flux A=the bond area over which the new mass is distribute Ω =the volume of a single atom or molecule The deposited or removed atoms change the neck size and shape.-> flux High temperatures promote faster mass transport and thereby contribute to faster neck growth. <ul><li>particle size </li></ul><ul><li>temperature </li></ul><ul><li>time </li></ul>
Shrinkage Shrinkage during initial stage sintering follows a kinetic law. The parameter B is exponentially dependent on temperature Surface area Surface-area-reduction kinetics of the sintering of 0.2μm Alumina at 750 ℃.
Intermediate Stage Simultaneous pore rounding Dinsification Grain growth The driving force is elimination of the remaining surface energy. pore structure Retarded grain growth & enhanced diffusion: temperature, microstructure Grain growth becomes increasingly active as the Pore structure collapses. The pinning effect of the pores diminishes as they Shrink and occupy less grain boundary area. -> coasening :Smaller grains aid densification Densification rate
Grain growth forces the grain boundary to be curved, leading to a progressive increase in grain boundary area as the grain boundary bows to sustain contact with a slower-moving pore. A critical condition is achieved where it is favorable for the boundary to break away from the pore. grain growth rate > pore mobility : isolation pore (slow densification by long-range volume diffusion) grain growth rate < pore mobility : continue to shrink (surface diffusion or evaporation-condensation)
Usually, the grain boundary mobility is much larger than the pore mobility, leading to pore drag, which reduces the rate of grain growth during final-stage sintering, as long as the pores remain attached to the grain boundaries . pores motion : surface diffusion grain boundary motion : temperature, grain size and grain boundary energy The conditions where pores remain attached to grain boundaries during final-stage sintering : F p =the force on a pore which varies with solid-vapor surface energy divided by the pore size F G =the force on the grain boundary, varies with the curvature of the grain boundary N=parameter, varies with the inverse square of the pore spacing
Rate of grain growth : K f =a geometric constant, relates the pore spacing and the grain boundary curvature (≈1) For the typical case of pore motion by surface diffusion, a relative coarsening to densification ratio Γ : D S =surface diffusivity D B =grain boundary diffusivity γ SS =grain boundary energy γ SV =solid-vapor surface energy Γ < 1 : full density
Doped MgO : the coarsening to assist in densification Doped ZrO2 : assists densification by inhibiting grain growth Rapid grain growth is observed once the Grain boundaries break away from the pores.
Coarsening vs Densification <ul><li>Second-phase particles and segregants slow grain growth. </li></ul><ul><li>-> Zener effect </li></ul><ul><li>Prolonged sintering -> Ostwald ripening </li></ul><ul><li>If the pore contains a trapped gas, influences both coarsening and densification. </li></ul><ul><li>-> depending on the gas species </li></ul>
Avoiding the conditions that give boundary separation from the pores is essential for high sintered densities.
DATA ANALYSIS Slope = volume diffusion control Time exponent exactly matches viscous flow control The more rapid sintering at the higher temperatures. Arrhenius plot Slope = 405kJ/mol;activation energy
The compacts sintered by a combination of Surface and volume diffusion.
SINTERING DIAGRAMS A sintering diagram proves useful in condensing and representing sintering behavior. Indicates the dominant transport mechanism This plot shows the neck size ratio versus isothermal sintering temperature for four hold times.
The sintering diagram based on density for submicrometer alumina with four time lines. A plot of fractional density versus temperature for isothermal sintering of 6 μ m tungsten with two time lines.